THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


/ 


/^, 


i/i^cL, 


^\Q^ 


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PHYSICAL 


CELESTIAL    MECHANICS. 


BY 


BENJAMIN    PEIRCE, 


riiUKISS     ritOFESSOR     OF     ASTRONOMY     AND     MATIIKM  VTK  S     1  \      IIAKVAIMI      ISlVEIiSlTV, 

AND     CONSULTING    ASTRONOMER    OF    THE    AMERICAN    El'llLMEKlS 

AND     NAUTICAL    ALMANAC. 


DEVELOTED    IN    EOUIt     SYSTEMS     OF 


ANALYTIC   MECHANICS,  CELESTIAL   MECHANICS,  POTENTLVL 
PHYSICS,  AND  ANALYTIC  MOPtPPIOLOGY. 


BOSTON: 
LITTLE,    BROWN    AND    COMPANY. 

1855. 


A    S  Y  S  T  E  Jil 


OP 


ANALYTIC    MECHANICS. 


BENJAMIN    PEIRCE 


PERKIXS    PROFESSOR    OF   ASTR0X03IT    AND    MATHEMATICS    IN   HARVARD    UNITERSITY, 

AND    CONSULTING    ASTRONOMER    OF    THE   AMERICAN   EPHEMERIS 

AND    NAUTICAL   ALMANAC. 


BOSTON: 

LITTLE,    BROWN    AND    COMPANY. 

18  5  5. 


Entered  according  to  Act  of  Congress  in  the  year  1855,  by 

LITTLE,   BROWN   AND   COMPANY, 

In  the   Clerk's   Office  of  the  District   Court  of  the  District  of  Massachusetts. 


CA:MlillIDGE: 
ALLEN     AND      F  A  K  N  11  A  M  ,      P  K  I  N  T  E  R  S  . 


Engineering  & 

Mathematical 

Sciences 

Library 


TO 


THE     CUKRISIIF.D    AND     REVERED     MEMORY     OF 


MY     :M  ASTER     IN     SCIENCE, 


X  A  T  11  A  N  I  E  L     B  0  W  D  I  T  C  II , 


THE      FATHER     OF      A3IERICAN      GEOMETRY, 


THIS     VOLUME 


IS     INSCRIBED. 


G21232 


ADVERTISEMENT. 


The  substance  of  the  present  volume  was  originally  pre- 
pared as  part  of  a  course  of  lectures  for  the  students  of  mathe- 
matics in  Harvard  College.  But  at  the  request  of  some  of  my 
pupils,  and  especially  of  my  friend  Mr.  J.  D.  Runkle,  I  have  been 
induced  to  undertake  its  publication.  The  liberality  of  my 
publishers,  the  well-known  firm  of  Little,  Brown  &  Co.,  who  gen- 
erously gave  directions  to  the  printers,  that  no  expense  should  be 
spared  in  its  typographical  execution,  seemed  to  impose  upon  me 
an  increased  obligation  to  perform  my  portion  of  the  task  to 
the  best  of  my  ability.  I  have  consequently  reexamined  the 
memoirs  of  the  great  geometers,  and  have  striven  to  consoli- 
date their  latest  researches  and  their  most  exalted  forms  of 
thouo-ht  into  a  consistent  and  uniform  treatise.  If  I  have, 
hereby,  succeeded  in  opening  to  the  students  of  my  country  a 
readier   access   to   these   choice  jewels   of  intellect,  if  their   bril- 


yjj.^  ADVERTISEMENT. 

liancy  is  not  impaired  in  this  attempt  to  reset  them,  if  in 
their  new  constellation  they  illustrate  each  other  and  concen- 
trate a  stronger  light  upon  the  names  of  their  discoverers,  and 
still  more,  if  any  gem  Avhich  I  may  have  presumed  to  add,  is 
not  wholly  lustreless  in  the  collection,  I  shall  feel  that  my 
work  has  not  been  in  vain.  The  treatise  is  not,  however, 
designed  to  be  a  mere  compilation.  The  attempt  has  been 
made  to  carry  back  the  fundamental  principles  of  the  science 
to  a  more  profound  and  central  origin  ;  and  thence  to  shorten 
the  path  to  the  most  fruitful  forms  of  research.  It  has, 
moreover,  been  my  chief  object  to  develop  the  special  forms 
of  analysis,  which  are  usually  neglected,  because  they  are  only 
applicable  to  particular  problems,  and  to  restore  them  to  their 
true  place  in  the  front  ranks  of  scientific  progress.  The 
methods  which,  on  account  of  their  apparent  generality,  *  have 
usually  attracted  the  almost  exclusive  attention  of  the  student, 
are,  on  the  contrary,  reestablished  in  their  true  position  as 
higher   forms   of  speciality. 

BENJAMIN  PEIRCE. 


—  67  — 
shell  is 

1.     For  an  internal  point   this  equation  becomes,  by  §§  135 
and  144, 


the  intearral  of  which  is 


/* 


M    ' 


to  which  no  constant  need  be  added,  because,  when  the  dimensions 
of  the  shell  are  infinite,  S2  and  i2^  both  vanish,  since  all  the  points 
of  action  are  infinitely  remote  from  the  centres  of  action.  This 
equation  expresses  that  the  potential  of  each  shell  has  the  same 
value  for  all  internal  points,  and,  therefore,  there  is  no  tendency  to 
motion  ^vithin  the  shell,  and  the  surface  of  the  shell  must  be  level, 
with  reference  to  its  own  action. 

2.     For   an   external   point,  the    equation  (G72)   becomes,   by 
§135, 

Hence,  by  integration, 

—  =  a  constant, 

which  constant,  however,  depends  for  its  value  upon  the  position  of 
the  points  of  action  ;  but  since  it  has  the  same  value  for  all  the 
shells  to  which  the  point  is  external,  the  potential  is  constant  for 
the  same  series  of  points  external  to  one  shell  for  which  it  is 
constant  through  the  action  of  another  shell ;  that  is,  all  the  shells 
have  the  same  external  level  surfaces.  But  the  external  level 
surface,  which  is  nearest  to   any  shell,  differs  infinitely  little  from 


—  68  — 

the  level  surface  of  the  shell  itself,  and,  therefore,  the  surface  of 
each  shell  is  a  level  surface  for  every  included  shell.  Hence,  the 
external  level  surfaces  of  a  shell  are  the  same  with  those  of  the 
original  masses,  and  the  attraction  of  a  shell  upon  an  external  point 
has  the  same  direction  with  the  attraction  of  the  original  masses, 
and  is  normal  to  the  level  surface  passing  through  the  point.  This 
theorem  is  due  to  Chasles. 

146.  Every  infinitely  thin  shell,  of  ivhich  the  surface  is  level,  from  the 
action  of  the  shell  itself,  must  he  a  Chaslesian  shell.  For,  if  another  shell 
is  constructed  upon  this  level  surface,  which  is  the  negative  of  the 
Chaslesian,  one,  namely,  which  is  repulsive,  instead  of  being  attrac- 
tive, or  the  reverse,  and  the  whole  mass  of  which  is  equal  to  that  of 
the  given  shell,  the  two  shells,  having  the  same  level  surfaces, 
exactly  cancel  each  other's  action  throughout  all  space.  The 
elements  of  mass  of  the  two  shells  must  then  be  absolutely  equal, 
but  of  opposite  signs  at  every  point.  For,  if  they  were  unequal  at 
any  point,  that  point  might  be  made  the  centre  of  an  infinitely  thin 
circular  element  of  the  combined  shells.  From  the  symmetry  of  its 
figure,  a  level  surface  for  the  action  of  this  element  alone  might  be 
made  to  pass  through  its  perimeter,  and  which  could  inclose  no 
other  mass  than  the  element  itself  But  such  surface  cannot  be 
level  for  the  remainder  of  the  combined  mass  of  the  two  shells,  and, 
therefore,  the  value  of  the  potential  upon  this  surface  for  the 
combined  masses  of  both  shells,  including  the  circular  element, 
cannot  be  constant.  This  want  of  constancy  in  the  potential  is 
contradicted  by  the  fact  that  the  shells  balance  each  other's  action 
everywhere.  There  cannot,  therefore,  be  any  such  want  of  con- 
stancy, nor  any  point  for  which  the  element  of  mass  of  the  given 
shell  is  not  absolutely  equal  to  that  of  the  Chaslesian  shell,  although 
it  is  of  a  contrary  sign.  But  reversal  of  the  sign  of  the  action  of 
the  mass  does  not  interfere  with  the  Chaslesian  characteristic  of  the 
shell. 


—  GO  — 

147.  Two  Chaslesian  shells,  v:Ucli  are  constructed  upon  the  same 
surface,  only  differ  in  their  densitf/  and  their  modulus  of  thickness.  For 
the  density  of  either  of  them  may  be  increased  or  decreased  mitil 
the  value  of  its  potential  at  the  common  surface  shall  be  equal  to 
that  of  the  other  shell.  If,  then,  its  action  be  reversed,  the  value 
of  the  potential  for  the  combined  shells  will  be  zero  both  at  the 
surface  and  at  an  infinite  distance  from  the  surface  ;  and  it  cannot 
have  any  other  value  in  the  intermediate  space,  otherwise,  there 
would  be  points  or  surfaces  of  maximum  potential  exterior  to  the 
acting  masses.  The  combined  surfaces  have,  therefore,  neither 
external  nor  internal  action,  and  the  reasoning  of  the  preceding 
article  demonstrates  that  the  component  shells  are  identical,  except 
in  reo:ard  to  their  siorns. 


ATTRACTION    OF    AX    ELLIPSOID. 


148.  An  infinitelf/  thin  homogeneous  shell,  of  which  the  inner  and 
Older  surfaces  are  those  of  sinular,  and  similarly  placed,  concentric  ellipsoids, 
is  a  Chaslesian  shell.  For,  if  upon  the  longest  axes  of  these  ellipsoids, 
as  diameters,  two  concentric  spheres  are  constructed,  each  sphere 
may  be  compressed  into  the  corresponding  ellipsoid,  by  reducing 
all  the  coordinates  from  the  centre,  as  origin,  parallel  to  either  of 
the  two  shorter  axes  of  the  ellipsoid  in  the  ratio  of  the  longest  axis 
to  this  shorter  axis.  But  all  points,  which  are  originally  in  the 
same  straight  line  remain  upon  a  common  straight  line  after  this 
imiform  compression  ;  and  all  distances  which  are  measured  in  the 
same  direction  are  reduced  in  a  common  ratio.  But  the  thick- 
nesses of  the  spherical  shell,  measured  upon  any  straight  line  at  the 
two  points  where  this  line  cuts  the  shell  are  equal ;  so  that  the 
thicknesses  of  the  ellipsoidal  shell,  measured  at  the  two  points 
where  the  reduced  line  cuts  this  shell,  are  also  equal.     If,  then,  at  a 


—  70  — 

point  assumed  at  will,  as  the  vertex,  within  the  ellipsoidal  shell,  an 
infinitesimal  cone  is  constructed  and  extended  in  each  direction 
from  the  vertex,  till  it  intersects  the  shell,  the  relative  masses  of  the 
two  included  portions  of  the  shell  are  proportional  to  the  squai^s  of 
their  distances  from  the  vertex;  and,  therefore,  their  attractions 
upon  the  vertex  are  equal,  but  in  opposite  directions.  Hence,  the 
action  of  any  portion  of  the  shell  upon  an  internal  point  is  balanced 
by  the  action  of  the  opposite  portion,  and  there  is,  consequently,  no 
tendency  to  motion  within  the  shell  from  its  own  action.  The 
surface  of  the  shell  is  thus  proved  to  be  a  level  surface,  in  respect  to 
its  own  action,  and,  by  §  140,  it  can  be  no  other  than  a  Chaslesian 
shell. 

149.  This  proposition  may  be  enlarged  to  a  theorem  given  by 
Newton,  for  a  finite  shell,  of  which  the  inner  and  outer  surfaces  are 
those  of  similar  and  similarly  placed  concentric  ellipsoids.  Such  a 
shell  may  be  called  a  Newtonian  shell,  so  that  the  infinitely  thin 
Newtonian  shell  is  a  Chaslesian  ellipsoidal  shell.  But  the  New- 
tonian shell  may  be  subdivided  by  similar  and  similarly  placed 
concentric  ellipsoidal  surfaces  into  an  infinite  number  of  Chaslesian 
ellipsoidal  shells,  each  of  which  is  inactive  with  reference  to  an 
internal  point.  Hence,  the  tvhole  Newtonian  shell  exerts  no  action  wjion 
an  internal imnt. 

150.  An  ellipsoid  may  be  converted  into  any  other  similar, 
and  similarly  placed,  concentric  ellipsoid  by  a  process  similar  to  that 
by  wdiich  the  sphere  in  §  148  was  changed  to  an  ellipsoid  ;  that  is, 
by  increasing  or  decreasing  the  coordinates  of  each  point,  taken 
from  the  centre  as  origin,  and  parallel  to  either  axis,  in  the  ratio  of 
the  corresponding  axes  of  the  two  ellipsoids.  The  points  of  the  two 
ellipsoids,  which  correspond  in  this  process,  have  been  called  by 
Ivory  corresponding  iwints.  By  this  process,  any  Newtonian  shell 
may  be   converted   into   another   concentric   and   similarly  placed 


LIST    OF    SUBSCRIBERS. 


J.  I.  Bowditch,  (10  copies), 

Boston. 

John  D,  Rimkle,  (5  copies), 

Cambridge. 

Chaimcey  Wright,  (2  copies). 

a 

C.  H.  Sprague,  (2  copies). 

Maiden. 

W.  C.  Kerr, 

Davidson  College,  X.  C 

George  Eastwood, 

Saxonville. 

Charles  PhilHps, 

Chapel  Hill,  X.  C. 

Joseph  W.  Sprague,  (2  copies). 

Rochester,  N.  Y. 

J.  M.  Chase, 

Cambridge. 

R.  H.  Chase, 

a 

Sharon  Tyndale, 

a 

Isaac  Bradford, 

a 

John  Bartlett,  (3  copies), 

a 

Gustavus  Hay, 

Boston. 

F.  J.  Child, 

Cambridge. 

William  C.  Bond,  for   Observatory    of 

Harvard  College,  (2  copies), 

a 

J.  E.  Oliver,  (2  copies), 

Lvnn. 

C.  W.  Little, 

Cambridge. 

N.  Hooper, 

Boston. 

C.  F.  Choate,  (2  copies), 

Cambridge. 

X 


LIST   OF   SUBSCRIBERS. 


J.  P.  Cooke,  Jr., 

B.  A.  Gould,  Jr., 
Joseph  Winlock, 
H.  L.  Eustis, 
Joseph  Lovering, 

C.  Gordon, 
Jared  Sparks, 
A.  Brown, 
WilHam  G.  Choate, 
J.  F.  Flagg,  Jr., 

A.  E.  Agassi  z, 

William  G.  Pearson, 

Charles  Sanders,  (2  copies), 

Theophilus  Parsons, 

John  Erving,  Jr., 

Charles  H.  Mills, 

Edmund  D wight, 

Edward  Everett, 

J.  H.  C.  Coffin, 

T.  S.  Hubbard, 

Mordecai  Yarnall, 

James  Major, 

James  Ferguson, 

R.  B.  Hamilton, 

Stej^hen  Alexander,  (2  copies), 

James  Walker, 

William  Chauvenet,  (4  copies), 

Washington  Observatory,  (5  copies), 

Thomas  Hill, 

Waltham  Rumford  Institute, 

Charles  Avery, 


Cambridge. 

(( 

a 
a 

u 

a 
a 

a 

Washington,  D.  C. 
Cambridge. 

a 
a 
a 

a 

Boston. 

a 

iC 

Annapolis,  Md. 
Washington,  D.  C. 


u 

a 

a 

a 

li 

a 

Syracuse,  N. 

Y. 

Princeton,  N 

.J. 

Cambridge. 

Annapolis,  Md. 

Washington, 

D.  C, 

Waltham. 

Hamilton  College,  N.  Y. 


LIST   OF   SUBSCRIBERS. 


XI 


John  Paterson, 

Albany,  N.  Y. 

Albany  State  Library, 

a              u 

State  Normal  School, 

a             a 

George  R  Perkins, 

Utica,  N.  Y. 

A.  D.  Bache,  (4  copies). 

Washington,  D.  C. 

American  Nautical  Almanac,  (5  copies), 

Cambridge. 

J.  W.  Jackson, 

Union  College,  N.  Y. 

Robert  S.  Avery, 

Washington,  D.  C. 

A.  W.  Smith, 

JVIiddletown,  Conn. 

J.  M.  Vanvleck,  (2  copies), 

a                    a 

Sarah  Watson, 

Nantucket. 

Smithsonian  Institution,  (25  copies), 

Washington,  D.  C. 

George  W.  Phillips, 

Salem. 

H.  Bailliere,  (3  copies), 

New  York  City. 

E.  S.  Snell, 

Amherst. 

John  F.  Frazer, 

Philadelphia. 

N.  Fisher  Longstreth, 

a 

Fairman  Rogers, 

a 

E.  Otis  Kendall, 

iC 

Ira  Young, 

Dartmouth  College. 

P.  H.  Sears, 

Boston. 

W.  F.  Phelps, 

Albany. 

Edward  L.  Force, 

Washington,  D.  C. 

Didactic  Society  of  University  of  N.  C, 

Chapel  Hill,  N.  C. 

F.  B.  Downes, 

Racine,  Wis. 

W.  G.  Peck, 

West  Point,  N.  Y. 

Elias  Loomis, 

New  York  City. 

John  Tatlock, 

Williamstown  College, 

Edward  Pearce, 

Providence,  R.  I. 

James  Mills  Peirce. 

Cambridoje. 

C.  W.  Eliot, 


Xll 


LIST   OF   SUBSCRIBERS. 


F.  W.  Bardwell, 

Cambridge. 

Charles  L.  Fisher, 

a 

William  Dearborn, 

Boston. 

Sidney  Coolidge, 

a 

L.  R.  Gibbs, 

Charleston,  S.  C. 

J.  D.  Crehore, 

Newton. 

J.  G.  Cogswell,  Astor  Library, 

New  York  City. 

Wolcott  Gibbs, 

a                 u 

Triibner  &  Co.,  (25  copies). 

London. 

William  S.  Haines, 

Providence,  R.  L 

Alexis  Caswell, 

a            « 

George  M.  Hunt, 

Stuyvesant,  N.  Y. 

Charles  C.  Snow, 

Brooklyn,  N.  Y. 

B.  Westermann, 

New  York  City. 

W.  H.  Cathcot, 

Urbanna  University,  Ohio, 

S.  C.  Huntington, 

Pulaski,  Orange  Co.,  N.  Y. 

Edward  King, 

New  York  City. 

A.  G.  Harlow, 

Cambridge. 

C.  A.  Cutter, 

a 

R  T.  Paine, 

a 

C.  F.  Sanger, 

a 

John  B.  Tileston, 

i( 

J.  M.  SewaU, 

li 

B.  S.  Lyman, 

a 

C.  H.  Davies, 

Cl 

Isaac  A.  Hagar, 

a 

F.  H.  Smith, 

Universitv  of  Yiro-inia. 

ANALYTICAL  TABLE  OF  CONTEXTS. 


CHAPTER    T. 

M  O  T  I  O  X ,     FORCE,      A  X  D      :M  A  T  T  K  R . 


Section 

1.  The  universality  of  motion,  . 

2.  The  spiritual  origin  of  force. 


Page    Section 

.     1  1    3.  The  inertia  of  matter, 
1 


Page 
.     1 


CHAPTER    II. 

MEASURE     OF     M  O  T  I  O  X     A  X  D      FORCE. 


I.   The   JiIeasure  of  Motiox. 

4.  Uniform  motion,  ....  2 

5.  Velocitj-  defined,     ....  2 

6.  Formula  of  varying  velocity  (227),       .  2 

n.   The  Measure  of  Force. 

7.  Force  and  power,        ....  2-3 

8.  The  force  is  proportional  to  the  ve- 

locitj-,     3 


9.  The  mass  defined,   . 
10.  The  inertia  of  a  body, 


3 

3-4 


III.   Force  of  jiovixg  Bodies. 

11.  The   power   is    proportional    to    the 

square  of  the  velocity,   ...       4 

12.  The  power  is  half  the  product  of  the 

force  by  the  velocity,      ...       4 

13.  Formula  of  varying  force,       .        .  5 


CHAPTER    III. 

FUXDAMEXTAL     PRIXCIPLES     OF     REST     AXD     M  O  T I O  X. 


I.   Tendency  to  Motiox*. 

14.  The  total  force  of  a  system  of  bodies 
is  the  exact  equivalent  of  the  sum 
of  its  components,  ....       5 

1.5.  The  fundamental  principle  of  equi- 
librium, .....       5 

16.  The  measure  of  the  tendency  to  any 

form  of  motion,       ....  5-6  , 

1  7.  Formula  for  the  measure  of  the  ten- 


dency to  a  proposed  motion,  ex- 
pressed by  virtual  velocities  (''lo),    6-7 

U.  The  Equations  of  Motion  and  Rest. 

18.  The  equation  of  motion  (813)  and  that 

of  rest  (8a,), 7-8 

19.  These  equations  can  be  decomposed 

into  as  many  partial  equations  as 
there  are  independent  elements,    .  8-9 


XIV 


ANALYTICAL   TABLE   OF   CONTENTS. 


CHAPTER    IV. 

ELEMENTS      OF      MOTION. 


I.    Motion  of  Translation. 

20.  A  point   has  three   independent  ele- 

ments of  motion,  from  which  any 
other  elementary  motion  can  be 
obtained  by  the  formula  (lO,:;),        9-10 

21.  Translation  and  rotation,  being  almost 

universal,  are  the  most  important 
forms  of  motion,  .         .         .  10-11 

22.  Definition  of  translation,     .         .         .11 

23.  Parallelojiiped  of  translation,  .         .         11 

II.   Motion  of  Rotation. 

24.  Definition  of  rotation,         .         .         .12 

25.  Projection  of  rotation  upon  any  axis,  12-13 

26.  27.  Parallelopijjed  and  parallelogram 

of  rotations,         ....  13-14 
28,  29.  Elementary    rotations    projected 

upon  a  given  direction,       .         .  14-15 
30,  31.  Theorem  of  two  systems  of  rectan- 
gular axes  (15»g),        .         .         .  15-16 

III.    Combined   Motions  of  Rotation  and 
Teanslation. 

32.  Decomposition  of  a  rotation  into  a  ro- 

tation about  a  parallel  axis  and 
a  translation  in  a  peri^endicular 
plane, 16 

33.  Combination  of  rotations   about  par- 

allel axes, 16 

34.  Equations  of  the   axis   of  combined 

parallel  rotations,        .         .         .  16-17 

35.  Combined  equal  rotations  about  jjar- 

allel  axes, 17 

36.  Combined    rotations   about    opposite 

axes, 17-18 

37.  Combined  rotations  which  are  equiva- 

lent to  a  translation,       .         .         .18 


38.  A  couple  of  rotations,     .         .         .         18 

39.  Combination  of  rotation  and  transla- 

tion,     18 

40.  Analysis  of  every  possible  motion  of 

a  solid,  into  a  screw  motion,        .  18-19 

41.  The  instantaneous  axis  of  rotation,     .     19 

42.  Special  mode  of  conceiving  the  motion 

of  a  solid  by  means  of  the  surfaces 
described  by  the  instantaneous 
axis,       .         .         .         .         .         .19 

43.  Case  in  which  the  surfaces  of  §42  are 

developable,        ....  19-20 

44.  Case  in  which  the  surfaces  of  §  42  are 

cylinders  or  cones,      .         .         .  20-21 

45.  General  case  reduced  to  that  of  §  44, 

combined  with  translation,  .         21 

46.  Axes  of  greatest  curvature  of  the  coni- 

cal surface,         ....         21 

47.  Decomposition   of   the    rotation   into 

rotations  about  the  axes  of  greatest 
curvature,       .         .         .         .         .22 

48.  Relations  of  the  velocities  of  rotation 

to  that  of  the  instantaneous  axis,  22-23 

49.  Relations  of  the  rotations  when   the 

surtaces  are  cylinders,    .         .         .23 


IV.    Special  Elements  of  Motion  aj«'D 
Equations  of  Condition. 

50.  The  independent  elements  of  position,     24 

51.  The  equation  of  condition  for  depend- 

ent elements,      ....  24-25 

52.  53.  Elimination  by  the  method  of  mul- 

tipliers,         25-26 

54.  The  variation  of  the  equation  of  con- 
dition expressed  by  means  of  the 
variation  of  the  normal  to  a  cor- 
responding surface,     .         .         .  26-27 


ANALYTICAL  TABLE  OF  CONTENTS. 


XV 


CHAPTER    V. 

F  O  1{  C  E  S     OF     NATURE. 


I.  The  Potential,  Level  Surfaces,  Posi- 
tions OF  Equilihrium,  and  the  Pos- 
sibility OF  Perpetual  Motion. 

55.  The  fixed  laws  are  not  incompatible 

with  the  spiritual  origin  of  force,    .     28 

56.  Fixed  and  variable  forces,       .         .         28 

57.  The  relation  of  the  forces  of  nature  to 

form  expressed  by  the  potential,    28-29 

58.  The  dependence  of  the  power  of  a 

system  upon  its  form,     .         .         .29 

59.  Limits  of  motion  of  a  system,  .         .         29 

60.  61.  The  potential  is  a  maximum  or  a 

minimum  for  the  position  of  equi- 
librium,         29-30 

62.  The  relation  of  stability  of  equilibrium 

to  the  maximum  or  minimum  of 
the  potential,  .... 

63.  There  are  as  many  positions  of  stable 

as  of  unstable  equilibrium,  with  ref- 
erence to  each  element  of  equi- 
librium, ...... 

64.  The  necessity  of  the  potential  in  the 

fixed  forces  of  nature,  and  its  rela- 
tion to  the  possibility  of  perpetual 
motion,  ...... 

65.  The  level  surface  and  its  finite  extent, 

66.  The  direction  of  attraction  is  perj^en- 

dicular  to  the  level  surface,    . 

67.  The  law  of  attraction  determined  by 

the  distance  apart  of  two  infinitely 
near  level  surfaces.   Level  surfaces 
do  not  intersect  each  other.     The  ' 
continuity  of  the  potential  of  nature,    32 

68.  The  trajectory  of  level  surfaces  termi- 

nates in  a  maximum  or  minimum,  32-33 

69.  The  limits    in  space  of  the  constant 

potential  coincide  with  those  of  the 
discontinuity  of  the  potential  or  Its 
derivatives, 33 

70.  There  is  no  force  or  mass  throughout 

a  space  of  constant  potential,  .     33 

71.  The  potential  of  nature  and  its  deriva- 


30 


30 


32 


33 


tlves  are  finite  and  continuous 
throughout  a  space  which  contains 
no  mass,  ..... 

72.  A  j^ortlon  of  sjiace,  for  which  the  po- 

tential of  the  fixed  forces  of  nature 
is  constant,  is  completely  bounded 
by  a  continuous  mass,  .         .  33-34 

73.  The  potential  of  nature  for  tempora- 

rily fixed  forces  may  vanish  for  an 
infinite  extent  of  space, 

74.  75.  The  computation  of  the  difference 

of  the  potential  for  two  points  by 
the  formula  (342rj), 


34 


34 


n. 


Composition  and  Resolution  of 
Forces. 


76. 


35 


35 


All  the  phenomena  of  nature  dejiend 

upon  combined  forces,    . 
The  projection  of  a  force  in  a  given 

direction,        ..... 
The  action  of  a  combination  of  forces 

in  any  direction,  .         .         .  35-36 

The  par  aUelopiped  of  forces^        .         .36 
8L  The  resultant  of  forces,  and  its  al- 
gebraic expression  (3 705),   .         .  36-37 
The  tendency  of  a  system  to  a  motion 

of  translation,  ....  37-38 
The  moment  of  a  force,  .  .  .38 
The  jjrojectlon  of  a  moment,  .  .  38-39 
The  par allelopiped  of  moments,  . 
The  moment  of  a  force  measures  its 

tendency  to  produce  rotation. 
The  positive  direction  of  the  axis  of  a 

moment,         ..... 
The  resultant  moment   measures  the 

total  tendency  to  produce  rotation. 
The  resultant  moment  of  forces  which 

act  upon  a  point  is  the  moment  of 

their  resultant,        .... 
9L  The  moments  for  parallel  lines, 
The    resultant   moment    ibr  different 

points,    ...... 


39 


39 


39 


40 


XVI 


ANALYTICAL  TABLE  OF  CONTENTS. 


93.  A  couple  of  forces,         ...         40 

94.  The  moment  of  a  couple  is  constant 

for  all  points  of  space,    .         .         .40 

95.  The  tendency  of  a  system  of  forces  to 

produce  translation  and  i-otation 
may  be  reduced  to  a  resultant  and 
a  resultant  couple,      .         .         .  40-41 

96.  It   may  be  still  further   reduced   to 

two  forces,      .         .         .         .         .41 

97.  The    resultant     and    the    resultant 

moment  may  always  coincide  in 
direction,        .         .         .         .         .41 

98.  99.  If  the  forces  arc  in  the  same  plane, 

or  if  they  are  parallel,  the  combined 
equivalent  is  either  a  resultant  or  a 
resultant  moment,  .         .         .42 

100.  Analytic  determination  of  the  com- 

mon direction  of  the  resultant  and 
resultant  moment  (489),      .         .  42-43 

101.  The  special  reduction  of  forces  re- 

quires special  forms  of  analysis,      .     43 

III.   Gravitatiox,  and  the  Force  of 
Statical  Electbicity. 

102.  Gravitation,  and  Its  elementary  po- 

tential,   43 

103.  Statical  electricity,  and  its  element- 

ary potential,      ....  43-44 

104.  Law  of  distribution  of  electricity,      .     44 

105.  Potential  of  gravitation  and  electric- 

ity (45g), 44-45 

106.  Laplace's  equation  for  the  determi- 

nation of  the  potential  (465),      .  45-46 

107.  The  law  of  attraction  of  gravitation 

or  electricity  (40,;),         .         .  .40 

108-112.  The  Attraction  of  an  Infi- 
nite Lamina,  ....  46-48 

108.  The  potentlalof  an  infinite  lamina,  46-47 

1 09.  The  level  surtaces  of  a  uniform  lamina,   47 
110-112.  The  attraction  of  a  uniform  la- 
mina (484)  and  (48io),         .         .  47-48 

113.  Poisson's  Modification  of  La- 
place's Equation  for  an  In- 
terior Point  (49o),         .         .  48-49 

114-124.  The  Attraction  of  an  In- 
finite Cylinder,    .        .        .  49-54 


114.  The  form  of  the  potential  of  an  in- 

finite cylinder  (5O3,), .         .         .  49-50 

115.  The  level  surfaces  of  an  infinite  cyl- 

inder,      50 

116.  Form  of  the  attraction  of  an  infinite 

cylinder  (oO^.i^),     .         .         .         .50 
117-120.  The  attraction  of  an  infinite  cyl- 
inder upon  a  distant  point  (51 29), 

(52i0, 50-52 

121-123.  The  attraction  of  a  circular  cyl- 
inder (53^,)'  (542),  (54;),  (54,,),'  52-54 

125,  126.  Relation  of  the  Poten- 
tial to  its  Parameter,        .  54-55 

127,  128.  Attraction  of  a  Finite 
Point  upon  a  Distant  Mass. 
The  Centre  of  Gravity,      .  55-50 

129-132.  The  Attraction  of  a  Spher- 
ical Shell  (56^,),  (57,9),  (5728), 
(58s), 56-58 

133-147.  The  Action  and  Reaction 
OF  A  Surface  or  infinitely 
THIN  Shell  of  finite  extent. 
The  Ciiaslesian  Shell,         .  58-69 

133.  The  total  action  of  a  surface  normal 

to  itself, 58-59 

134.  That  of  a  plane,        .         .         .         .59 

135.  Gauss's  theorem  relative  to  the  angle 

subtended  by  a  surface  (60^4),      .  59-60 

136.  Gauss's  and  Chasles's  </ieo?-e?n  uj)- 
.  on  the  normal  action  of  masses  up- 
on a  surface  (61,5),     .         .         .  60-61 

137.  Any  level  surface  must  inclose  masses 

of  matter,  .....  61-62 

138.  The   potential    of  a   closed   sui'face 

which  is  level  to  itself  is  constant 
for  the  inclosed  space,    .         .         .62 

139.  The  maximum  limit  of  the  potential 

is  within  the  mass, .         .         .         .62 

140.  In  a  gravitating  system  there  is  no 

point  of  minimum  potential,   .         .     62 

141.  The  attraction  is  constant  upon  all 

the  sections  of  a  trajectory  canal 

by  a  level  surface,       .         .         •  02-03 

142.  Condensed  view  of  the  laws  of  attrac- 

tion, and  their  relation  to  the  pro- 
pagation of  heat,  .         .         .  03-64 


ANALYTICAL  TABLE  OF  CONTENTS. 


XVll 


143.  Correspondence   of  the    Chaslesian 

shells  upon  different  level  sur- 
faces,    64-65 

144.  The  ratio  of  the  mass  of  a  Chaslesian 

shell  to  the  inclosed  mass,       .         .     65 

145.  The  law  of  attraction  of  a  Chaslesian 

shell  (67g),  (6  723),        .         .         .  65-68 

146.  The  surface  which  is  level  to  itself  is 

a  Chaslesian  shell,  .         .         .68 

147.  Chaslesian  shells  upon  the  same  sur- 

face are  similar,      .         .         .         .69 

148-177.    The     Attraction     op     an 

Ellipsoid,        ....  69-88 

148.  The  Chaslesian  ellipsoidal  shell,       69-70 

149.  The  Newtonian  shell,    ...         70 

150.  Ivory's    corresponding     points    of 

ellipsoids, 70-71 

151.  The  corresponding  elements  of  New- 

tonian shells  are  proportional  to 
the  shells, 71-72 

152.  Homofocal  Ne^vtonian  shells,    .         .     72 

153.  Corresponding  projections  of  corre- 

sponding radii  vectores  of  Newton- 
ian shells, 72-73 

154.  Diffei'ence  of  the  squares  of  corre- 

sjjonding  radii  vectores,  .         .73 

155.  Distance  of  corresponding  points,  .  73-74 

156.  The  external  level  surfaces  of  ellip- 

soidal Chaslesian  shells,  .         .     74 

157.  The  attractions  of  homofocal  Newto- 

nian shells, .....  74:— lb 

158.  The  attraction  of  a  Chaslesian  shell 

upon  a  point  of  its  surface  (7633),  75-76 

159.  The  attraction  of  a  Chaslesian  shell 

upon  an  external  point  (763),     .  76-77 

160.  There  are  three  surfaces  of  the  second 

degree  which  pass  through  a  given 
point,  and  have  given  foci,  of  which 
one  is  ellipsoidal,  and  the  other  two 
are  hy  perboloids  of  different  classes,     7  7 

161.  The  common  intersection  of  the  two 

hyperboloids  cuts  all  homofocal 
ellipsoids  in  corresponding  points,  77-78 

162.  The  three  surfaces  of  §  60  cut  each 

other  rectangularly,    .         .         .   78-79 

163.  The  common  intersection  of  the  two 

hyperboloids  is  a  transversal  to  the 


level  ellipsoidal  surfaces  of  the 
same  foci,       .         .         .         .         .79 

164.  Dupin's  theorem  that  orthogonal  sur- 

faces cut  each  other  in  the  lines  of 
greatest  and  least  cui~oatnre,         .  79-80 

165.  Hyperboloidal  Chaslesian  shells,       .     80 

166.  167.  The    attraction  of  an   ellipsoid 

on  an  external  point  (8234),  (83o),  80-83 

168.  Legendre's  formula  for  this  attrac- 

tion (83io), 83 

169.  The  expression  of  the  attraction  by 

elliptic  integrals  (84i8),  (8518.04),   83-85 

1 70.  Analytical  theorems  with  reference 

to  the  attractions  (863.17),   .         .  85-86 

171.  The  attractions  expressed  as  deriva- 

tives of  a  single  function  (8605.28),  •     86 

172.  The  equation  for  limits  of  integra- 

tion (87„), 87 

1 73.  The  condition  that  the  attracted  point 

is  upon  the  surface  of  the  ellipsoid,     87 

174.  The    case    in   which  the    attracted 

point  is  within  the  ellipsoid,   .         .87 

175.  The  attraction  when  the  density  of 

the  ellipsoid  varies  so  that  the  com- 
ponent Chaslesian  shells  are  homo- 
geneous,          87 

176.  The  attraction  of  a  homogeneous  ob- 

late ellij^soid  of  revolution,  .  87-88 

177.  The  attraction  of  a  homogeneous  jiro- 

late  ellijisoid  of  revolution,     .         .     88 

178-218.  The  Attraction  op  a  Sphe- 
roid. Legendre's  and  La- 
place's Functions,      .        .  88-116 

178.  Jacobi's  method  adopted  in  the  in- 

vestigation of  the  Legendre  a7id 
JjAVLACE  functions,  .  .  .88 
179-183.  Investigation  of  the  fundamental 
equations  (89is),  (gOg),  (dO..-;),(dOsi), 
for  the  determination  of  the  ele- 
ments of  these  functions,    .         .  89-90 

184,  186,  187.  The   relation  of  the   suc- 

cessive elementary  coefficients,  and 
their  derivatives  (912^,),  (9213-93.4), 

91-93 

185.  The  Eulerian  Gamma  integral.     See 

720te,  page  35G,    ....  91-92 
188,  189.  The    elementary    functions    of 


XVlll 


ANALYTICAL   TABLE   OF   CONTENTS. 


Legendre  AvMcli  vanish  (OSai), 
(94,), 93-94 

190.  The  general  vahies  of  the  elementary 

functions  for  positive  powers  (94o.,),     94 

191.  The    elementary   functions    for   the 

power  of  negative  unity  (953o),  .  94-95 

192.  The  elementary  functions  for  nega- 

tive powers  (974),       .         .         .  95-97 

193.  Development  of  the  distance  of  two 

points  according  to  the  powers  of 
their  radii  vectores,    .         .         .  97-98 

194.  The   values  of  Legendre's  or  La- 

place's functions  in  this  develop- 
ment (993),  ....  98-99 

195.  201.  Development   of   the   potential 

of  the  sjiherold,     .         .        99,102-103 

196.  Generalformof  these  functions  (IQQ^,  100 

197.  Poisson's  theorem  for   these   func- 

tions,          100-101 

198.  General  theorem  for  development  hy 

these  functions,        .         .         .         .101 

199.  200.  Laplace's  theorems  upon  these 

functions,       ....     101-102 

202.  Develojiment  of  the  potential  of  the 

sjiheroid  for  an  external  point 
when  the  origin  is  the  centre  of 
gravity, 

203.  The   homogeneous    ellipsoid    which 

coincides  in  the  first  two  terms  of 
the  development,  .         .         .     103- 

204.  Development  of  the  potential  for  an 

external  point  which  is  external  to 
the  spheroid  as  well  as  to  the  cor- 
responding homogeneous  ellipsoid 

205.  The  determination  of  the  axes  of  the 

corresponding  homogeneous  ellip- 
soid,           107-108 

20G-209.  Determination  of  the  potential 
for  a  point  which  is  quite  close  to 
the  spheroid  in  Poisson's  form  of 
analysis,        ....     108-112 

210.  The  attraction  of  a  spheroid  in  the 

direction  of  the  radius  vector,         .112 

211.  The  potential  of  the    homogeneous 

spheroid    for    an    external    point, 

112-113 

212.  The   potential  of  the   homogeneous 

spheroid  for  a  point  of  its  surface,  113 


103 


-10; 


107 


213.  Potential  of  the  spheroid  which  dif- 

fers little  from  the  ellipsoid,  .         .113 

214,  215,  216.    Potential    of   a   sjAeroid 

which  is  nearly  a  sphere,       .     113-115 

217.  Potential   of  a   spheroid  for   an  in- 

terior point, 116 

218.  The  discussion  of  the   convergence 

of  the  series  referred  to  subsequent 
volumes, 116 

IV.   Elasticity. 

219.  Nature  of  the  phenomena  of  elastic- 

ity,             116-117 

220.  Linear   expansion  of  a   body,   and 

elhpsoid  of  expansion,  .         .     117-118 

221.  Principal  axes  of  expansion,      .         .118 

222.  Surface  of  distorted  expansion,    118-119 

223.  Surface  of  distorted   expansion  re- 

ferred to  principal  axes,         .         .119 

224.  225,  226.  Rotative  effects  of  expan- 

sion,           119-120 

227.  Total  expansion  of  a  body,        .         .120 

228.  Linear  expansion  for  small  disturb- 

ance,          120-121 

229.  Reciprocal   expansive    ellipsoid  for 

small  disturbance, .         .         .         .121 

230.  Reciprocal    expansive    ellipsoid   re- 

ferred to  jirincipal  axes,         .         .121 

231.  Case  in  which  the  reciprocal  exjjan- 

sive  ellijjsoid  becomes  hyperboloid- 

al  or  cyhndrical,   .         .         .     121-122 

232.  Total  expansion   for  small  disturb- 

ance,       122 

233.  Rotation  for  small  disturbance,      .       122 

234.  Directions  of  maximum,   minimum, 

and  mean  rotation,        .         .     122-123 

235.  Combination  of  mean  rotations,        .  123 

236.  Rotation  for   small   disturbance   re- 

ferred to  principal  axes,         .         .123 

237.  Compression  without  mean  rotation,  123 

238.  Rotation  Avithout  compression,  .  123-124 

239.  The  discussion  of  the    elastic  force 

reserved  for  special  chapter, .         .  124 

V.   MoDiFYixG  Forces. 

240.  Modifying   forces    defined,   and    di- 

vided   Into    stationary   and    mov- 
ing,          124 


ANALYTICAL  TABLE  OF  CONTENTS. 


xiy 


241.  Relation     of    stationary    modifying 
forces   to   equations   of  condition, 


242.  Mode  of  action  of  modifying   force, 

125-126 


124-125    243.  Moving  modifying  forces, 


.  126 


CHAPTER    VI. 

EQriLIBRIUJI     OF     TRANSLATIOX. 

244.  The    conditions    of   equilibrium    of         I  lation  when  there  is  a  fixed  sur- 

translation  (127i4),         .         .         .127'  face,  line,  or  point,  .         .         .128 

245.  The    conditions    of    equilibrium    of  247.  The  equilibrium  of  a  material  point 

translation  are  the  same  as  if  all  wholly  included  in  translation,        .  128 

the  forces  were  applied  at  a  single  248.  In  the  equilibrium  of  translation  each 

point, 127-128  force  is  equal  and  opposite  to  the 

246.  Conditions  of  equilibrium  of  trans-  resultant  of  all  the  others,        .  128-129 


CHAPTER    VII. 

EQUILIBRIUM      OF     ROTATION. 

249.  The    conditions    of   equilibrium    of         1  253.  Equilibrium  of  rotation  when  there 

rotation, 129  are  two  fixed  points,       .         .         .130 

250.  When  the  equilibrium  of  rotation  is  254.  Relation  of  the  centre  of  grarity  to 

universal  that  of  translation  is  in-                        equilibrium  of  rotation  of  parallel 
Tolved, 129-130  I  forces, 130-131 


251.  Equilibrium  of  rotation  about  parallel 

axes,       .         .         .         .         .         .130 

252.  Axis  for  which  the  resultant  moment 

vanishes, 130 


255.  Internal  forces  neglected  in  the  con- 
ditions of  equilibrium  of  transla- 
tion or  rotation,  or  action  and  re- 
action, are  equal,    .        .         •  131-»132 


CHAPTER    VIII. 

EQUILIBRIUM     OF    EQUAL     AND    PARALLEL    FORCES. 


I.  JIaxima  axd  ]\Iixima  of  the  Potential.      |  259. 

256.  Gravitation  taken  as  the  tv-pe  of  these 

forces.    The  level  surfaces  are  hor-  i 

izontal  planes,        .         .         ...  132    260. 

257.  Relation  of  the  maximum  and  mini- 

mum potential  to  equilibrium,   132-133 

258.  The   equilibrium    of    translation    of  261. 

gravitation  requires  stationary  mod- 
ifying forces, 133  I 


The  resultant  moment  of  gi-avity 
vanishes  for  the  centre  of  grav- 
ity,   133 

Position  and  magnitude  of  a  single 
modifying  force  In  a  gravitating 
system,  ......  1^3 

Position  and  magnitude  of  two  mod- 
ifying forces  in  a  gravitating  sys- 
tem,        133 


XX 


ANALYTICAL  TABLE  OF  CONTENTS. 


262.  The  change  of  the  intensity  of  grav- 
ity does  not  affect  the  position  of 
equilibrium,    .         .         .         .         .134 


263 

264, 


265, 
266, 


267. 

268. 

269. 

270. 
271. 

272. 
273. 

274. 

275. 

276. 

277. 

278. 
279. 

280. 

281. 

282. 
283. 


n.  The  Funicular  and  the  Catenary. 

Tlie  funicular  defined,      .        .         .134 
The  funicular  with  one  fixed  point, 

134-135 

The  funicular  with  two  fixed  points,  135 

The  point  of  meeting  of  the  lines  of 
extreme  tension  of  any  portion  of 
the  funicular,  .         .         .         .  135-136 

The  vertical  projections  of  the  ex- 
treme tensions  with  reference  to 
the  distance  from  the  centre  of 
gravity, 136 

The  inclination  of  the  funicular  to 
the  horizon,     ....  136-137 

Point  of  change  in  the  funicular 
from  ascent  to  descent,    .         .  137-138 

The  equation  of  the  funicular  (138,4),  138 

The  general  equation  of  the  cate- 
nary (13831), 138 

The  catenary  of  uniform  chord  (1 3  93) ,  139 

The  uniform  chord  referred  to  rec- 
tangular coordinates  (13920-24),        •  139 

The  tension  of  the  uniform  chord 
(139.31), 139 

The  tension  and  thickness  of  a  cate- 
nary of  given  form  (1404_,o),  .         .  140 

The  catenary  of  uniform  strength 
(140,0-0;), 140 

The  catenary  for  uniform  support  of 
weight  (1412^),       .         .         .  140-141 

The  elastic  catenary  (Uljo-w),  .141 

The  catenary  upon  a  given  surface 
(142;^0» 141-142 

The  pressure  of  a  catenary  upon  a 
surface  (14 22g),       .       ' '.         .         .  142 

The  point  at  which  the  curvature  of 
the  catenary  upon  a  surface  van- 
ishes (143;),    ....  142-143 

The  catenary  upon  a  vertical  cylin- 
der (143i3), 143 

The  catenary  upon  a  vertical  surface 
of  revolution  (143jc,),      .         .         .   143 


284.  The  case  of  a  horizontal  catenary 

upon  a  surface  of  revolution  (14328), 

143-144 

285.  Direction  of  the  catenary  upon  a  sur- 

face of  revolution  (1 4430),       .        .  144 

286.  The  catenary  upon  the  vertical  right 

cone  with  a  circular  base  ;  its  equa- 
tion (145i2),  and  analysis  into  dis- 
tinct portions,  .         .         .  144-146 

287.  The  finite  portion  of  the  catenary 

upon  the  vertical  right  cone  ex- 
pressed by  elliptic  integrals  (14  7^), 

146-147 

288.  The  general  expression  of  the  arc  of 

the  spherical  ellipse  by  elliptic  in- 
tegrals (1492c),       .         .         .    147-149 

289.  The  expression  of  the  catenary  upon 

the  vertical  right  cone  by  the  arc 

of  the  spherical  ellipse  (ISO,),  149-150 

290.  Cases  In  which   the   catenary  upon 

the  vertical  right  cone  returns  into 
itself, 

291.  The  infinite  portions  of  the  catenary 

upon  the  vertical  right  cone  ex- 
pressed by  elliptic  integrals  (15O31), 
(I5I4), 150- 

292.  The  finite  and  infinite  portions  of  the 

catenary  may  be  expressed  by  the 
aid  of  reciprocal  sjiherlcal  ellipses, 

(151l7,2l),  .... 

293.  Case  in  which  the  finite   portion  is 

circular  (ISloj-ji),  . 

294.  Case  in  which  the  catenary  upon  the 

vertical  cone  degenerates  into  a 
straight  line,  .... 

295.  The  Investigation  of  the  infinite  por- 

tion of  the  catenary  upon  the  verti- 
cal right  cone,  when  the  finite  por- 
tion disappears  (1523i),  (153;),  152-153 

296.  Case,  recognized  by  Bobillier,  in    _ 

which  the  catenary  upon  the  verti- 
cal right  cone  becomes  an  equi- 
lateral hyperbola  upon  the  devel- 
oped cone  (15325), 

297.  The  catenary  upon  a  vertical  ellip- 

soid of  revolution,  its  equation 
(154,0), 

298.  The   cases  In  which  there  are  two 


150 


-151 


151 


151 


152 


153 


154 


ANALYTICAL  TABLE  OF  CONTENTS. 


XXI 


portions  of  the  catenary  upon  the 
vertical  ellipsoid  of  revolution,  or 
one  portion,  or  when  there  is  no 
catenary,        ....    154-155 

299.  The  cases  in  Tvhich  the  two  portions 

of  the  catenary  upon  the  vertical 
ellipsoid  of  revolution  are  similar 
(15635),  (157g_2g).  This  case  was 
recognized  by  Bobilliek  for  the 
sphere,  155-157 

300.  Expression  of  the  constants  by  means 

of  the  limiting  values  in  the  general 
expression  of  the  catenary  upon 
the  vertical  ellipsoid  of  revolution, 

157-159 

301.  Integral  of  the  differential  equation 


of  the  catenary  in  the  general  case 
of  the  vertical  surface  of  revolu- 
tion (loDo,), 159 

302.  The    catenary    upon     the    vertical 

equilateral  asymptotic  hj-perbolold 
when  the  incUnation  to  the  merid- 
ian is  constant,        .         .         .  159-160 

303.  The  definition  of  the  catenary  upon 

any  vertical  surface  of  revolution 
by  means  of  the  equilateral  asymp- 
totic hyperbolold,  ....  160 

304.  The  general  case   of  the   catenary 

upon  the  equilateral  or  asymptotic 
hyperbolold  (161 10),        .         .  IGO-lGl 

305.  The  limiting  point  at  which  the  cate- 

nary tends  to  leave  the  surface,     .  161 


CHAPTER    IX. 

ACTION     OF     MOVING     BODIES. 


306,  307.  Characteristic   Fuxctiox, 

162-163 

306.  Maupertius's  action  of  the  system 

and     Hamilton's     characteristic 
functions  (162.,i),         .         .         .162 

307.  Expenditure  of  action   hj  a  moving 

system  (1630),         .         .         .     162-163 

308.  Principle  of  Living  Forces  or 

Law  of  Power,  (163i4)         .  163 

309-312.  Canonical  Forms  of  the 
differential  Equations  of 
Motion,         .         .         .         .163-166 

309.  Lagrange's     canonical  forms    of 

equations  of  motion  (164io),         163-164 

310.  Equations  of  motion  expressed  in  rec- 

tangular coordinates  (I6403),      164,  165 
311, 312.  Hamilton's  modifications  of  La- 
grange's  canonical  forms  (I65..7), 
(I663), 165-166 

313-315.  Variations  of   the   Char- 
acteristic Function,  166-167 
313.  Derivatives  of  the  characteristic  func- 


tion with  reference  to  the  elements  of 
motion,         ....       166-167 

314.  Derivatives     of    the     characteristic 

function  for   the   rectangular  ele- 
ments of  motion,  .         .         .        167 

315.  Hamilton's    method    not   applicable 

when  the  forces  involve  the  veloc- 
ity,          167 

316-318.  Principle  of  Least  Action, 

167-169 

316.  Demonstration  of  the  principle  of  least 

action,         ....        167—168 

317.  Maupertius's  a  priori  deduction  of 

the  principle  of  least  action,         .      168 

318.  Deduction  of  the   dynamical  equa- 

tions from  the   principle    of  least 
action,         ....        168-169 

319-322.  Principal     Function     and 

other  similar  functions,  169-170 

319.  Hamilton's  principal  function  and 

its  use  (I699),  •         •         •         -169 

320-322.  Other  functions  suggested    by 


XXll 


ANALYTICAL  TABLE   OF   CONTENTS. 


Hamilton,  instead  of  tlie  charac- 
teristic functioa  (IGOoa),  (170i3, 24), 

169-170 

323-325.  Partial  Differential  Equa- 


tions FOR  THE  Determination 
OF  THE  Characteristic,  Prin- 
cipal, AND   other   Functions 

OF  THE  SAME  ClASS  (ITljg,,,), 
(17l27-172i),   ....  171-172 


CHAPTER    X, 


INTEGRATION  OF  THE  DIFFERENTIAL  EQUATIONS  OF  MOTION. 


32G.  Jacobi's  discussion  of  differential 
equations  important  to  a  full  de- 
velopment of  the  prohlem  of  me- 
chanics, .         .         .         .         .172 


I.    Determinants    and    Functional.    De- 
terminants. 

327.  Gauss's  determinants  (173i2),  .    172-173 

328.  Reversal  of  tlie  sign  of  the  determi- 

nant,       173 

329.  The  equaUty  between  the  elements 

for  which  the  determinant  van- 
ishes (173.V,),  ....  173-174 

330.  Reduction  of  the  forms  of  the  deter- 

minant when  certain  elements  van- 
ish (174i4,2o,23),        •         •         •         -174 

331.  An  element  can  be  taken  out  as  a 

factor  when  all  the  elements  of  a 
certain  class  vanish  (1742o,  gj),         .  174 

332.  Two  elements  can  be  taken  out  as  a 

factor,  by  an  extension  of  the  pre- 
ceding principle  (1753  5),        •         .175 

333.  By   the   ultimate    extension   of  this 

principle,  the  determinant  is  re- 
duced to  the  continued  product  of 
its  leading  terms  (175io,  in),     .         .175 

334.  The  complete  determinant  expressed 

by  means  of  the  partial  determi- 
nants (17521,07),       .         .         .         .175 

335.  Deduction  of  all  the  partial  determi- 

nants from  one,       .         .         .1 75-1 76 

336.  Conditional   equations  for  the   par- 

tial determinants  (13610,13),     .         .  176 

337.  Expressions  of  the  determinant  and 

of  the  partial  determinant  by  par- 


tial determinants  of  the  second  or- 
der (1 36.^,27),  .         .         •         -176 

338.  Mutual  relations  of  the  partial  deter- 

minants of  the  second  order,  and 
corresponding  reduction  of  the  ex- 
pression of  the  complete  determi- 
nant (177i2,i5),         .         .         .  176-177 

339.  The  solution  of  linear  equations  by 

the  aid  of  determinants  (17 701, 31),  .  177 

340.  Ratios   of   the    unknown  quantities 

when  the  second  members  vanish 
(17812), 178 

341.  342.  Determinants   found   from   the 

partial  determinants  taken  as  ele- 
ments (17931),  (180g),     .         .  178-180 

343.  The  variation  of  a  function  of  the 

elements,  and  of  the  determinant 

(180i3,25), 180 

344.  The  variation  in  a  special   case  of 

symmetrical  elements  (18030,31), 
(I8I0), 180-181 

345.  Inverse  solution  of  equation,  with  the 

corresponding  variations  (181ii_io),  181 

346.  Determinant  of  compound  functions 

of  two  systems  of  elements  (181 03), 
(182i2), 181-182 

347.  Cases  in  which  the  number  of  com- 

pound elements  is  not  more  than 
equal  to  that  of  the  simple  elements 
of  each  system  (182i8, 25),        .         .  182 

348.  Case  in  which  the  compounded  sys- 

tems are  identical  (1833,0),   .     182-183 

349-373.  Functional  Determinants, 

183-198 

349.  Definition  of  functional  determinant, 


ANALYTICAL   TABLE   OF   CONTENTS. 


XXUl 


183 


183 


184 


184 


185 


185 


and  its  relation  to  previous  propo- 
sitions (183o„),        .... 

350.  Case  in  wliieli  the  functional  deter- 

minant is  the  product  of  two  de- 
terminants (1833,), 

351.  Case  In  -which  the  functional  deter- 

minant is  a  continued   product  of 
derivatives  (I845), 

352.  The   determinant    of    mutually   de- 

pendent functions  vanishes,    . 

353.  A  dependent  function  is  constant  in 

finding  the  functional  determinant, 

184-185 

354.  Simplification  of  the  functional  deter- 

minant by  successive  substitution,  . 

355.  The  functional  determinant  of  inde- 

pendent functions  does  not  vanish, 

356.  The  functional  determinant  of  com- 

pound functions,      .         .         .  185-186 

357.  The    inverse    and   direct    function- 

al    determinants     are    reciprocals 
(187o,i„),  ....  186-187 

358.  Relation  of  inverse  derivative  to  par- 

tial functional  determinant  (187.,,),  187 

359.  Variation  of  functional  determinant 

(I885), 187-188 

360.  Variation  of  inverse  functional   de- 

terminant (188,j),   ....  188 

361.  Mutual  relation  of  the  partial  func- 

tional determinants  and  variations 
ofthe  functions  (I8823),  .         .188 

362.  Transformed  expression  of  the  func- 

tional determinant  (189,),       .  188-189 

363.  364.  Functional  determinant  of  im- 

plicit functions  (189.,,),  (190io),  189-190 

365.  Determinant  of  partial  functional  de- 

terminants (I9I5),  .         .         .  190-191 

366.  Determinant  of  pai'tial  functional  de- 

terminants of  inferior  order  (191,5),  191 

367.  368.  Determinants  of  mixed  partial 

functional  determinants   (192io, h), 

191-192 
369,  370.  Sum  of  products  of  mixed  par- 
tial determinants  (193^,21),      .  192-193 

371.  Lagrange's  equations  for  determi- 

nant of  partial  derivatives  (194o), 

193-194 

372.  Substitution  of  the  functional  deter- 


minant of  the  function  or  its  higher 
derivatives  for  the  first  derivative 
(194;,,), 194-195 

373.  Determination  of  a  system  of  func- 

tions for  which  the  functional  de- 
terminant is  given  (192i9_27), .         .  195 

374-375.  Multiple  Derivatives  and 

Integrals,   ....  196-198 

374.  Transformation  of  multiple   deriva- 

tives from  one  set  of  variables  to 
another  by  means  of  determinants 
(197,„), 196-197 

375.  Multiple  integrals  transposed  Into  a 

sum  of  linear  integrals  (198,5),  197-198 


n.    Simultaneous    Differential    Equations, 

AND     LiNEAK     PaeTIAL     DIFFERENTIAL 

Equations  of  the   First   Order. 

376.  An  Integral  of  simultaneous  differen- 

tial equations  (199,,,),     .         .         .199 

377.  The  solution  of  a  linear  partial  dif- 

ferential equation  (200.),        .  199-200 

378.  Relation  of  the  integral  of  simulta- 

neous differential  equations  to  the 
solution  of  a  linear  partial  differen- 
tial ecjuation,  ....  200 

379.  Transformation  of  linear  partial  dif- 

ferential equations  so  as  to  reduce 
the  number  of  variables,         .         .  200 

380.  A  solution  can  always  be  obtained 

by  series  (20I.2,),    .        .        .  200-201 

381.  The   number   of  independent   solu- 

tions of  a  linear  partial  differential 
equation,         ....  201-203 

382.  Any  function  of  the  solutions  is  a  so- 

lution,     203 

383.  General  and   particular  systems   of 

Integral  equations  (20325,31)?  • 

384.  Each  equation  of  a  general  system  of 

Integi-al  equations  Is  an  Integral,    . 

385.  Relations  of  a  particular  system  to 

the  integrals,  .... 

386.  That  portion  of  the  equations  of  a 

particular  system  which  does  not 
involve  arbitrary  constants  is  itself 
a  })articular  system. 


203 


204 


204 


204 


XXIV 


ANALYTICAL  TABLE  OF  CONTENTS. 


206 


206 


387,  The  conditional  equations  to  -whicli 
the  arbitrary  constants  of  a  general 
system  must  be  subject,  for  a  par- 
ticular system  (20831),     .         .  204-206 

888.  Process  of  deriving  a  system  of  in- 
tegral equations  from  one  of  its 
components,  ..... 

389.  Test  that  a  proposed  equation  is  not 
an  integral,    ..... 

890.  Superfluous  constants  lead   to    new 

integral  equations  (207,3),      .    206-208 

391.  The  system  of  integral  equations  in 

which  the  initial  values  are  the  ar- 
bitrary constants,  ....  208 

392.  An  integral  equation  in  which  the 

initial  values  are  the  arbitrary 
constants  may  be  changed  to  an- 
other integral  equation  in  which 
the  initial  values  are  the  varia- 
bles,          208-209 

393.  The  integral  equation  which  expresses 

the  value  of  an  initial  value  of  the 
variable,  transformed  to  one  which 
expresses  the  value  of  the  varialjle 
(209i3,»8), 209 

394.  DiiFerential  equations  of  high  orders 

reduced   to  the   first  order  when 
they  are  given  in  the  normal  form 
(210„.iO,         ....  209-210 
895.  Reduction  of  differential  equations 

to  the  normal  form,         .  •      .  210-211 

396.  Case  in  which  the  order  of  differen- 

tial equations  admits  of  reduction 

211-212 

397.  One  normal  system  transformed  to 

another, 212-213 

898.  Normal  systems  transformed  so  as  to 

contain  only  two  variables,     .  213-214 

399-431.  The  Jacobiax  Multiplier 
OF   Differential    Equations, 

214-231 

399.  Definition  of  Jacobian  multiplier,    .     214 

400.  The  functions  of  the  multijjlier  are 

the  independent  solutions,      .         .  214 

401.  The  linear  partial  differential  equa- 

tions by  which  the  multiplier  is  de- 
fined (215i2), ....  214-215 


402.  The   common    differential    equation 

which  defines  the  multiplier  (21 62), 

215-216 

403.  The  multiplier  expressed  by  a  deter- 

minant (21 6,1),       .         .         .         .216 

404.  The  multiplier  expressed  by  a  deter- 

minant of  implicit  functions  (2I617),  216 

405.  The  multiplier  expressed  by  the  in- 

verse determinant  (216:5,,),       .  216-217 

406.  The  ratio  of  two  multipliers  is  a  so- 

lution (217i5),         .         .         .         .217 

407.  Case  in  which  unity  is  a  multiplier 

(217.j), 217-218 

408.  Determination  of  the  multiplier  when 

its  solutions  are  known,  .         .218 

409.  Determination  of  the  multiplier  when 

the  solutions  have  the  forms  of  the 
initial  values  of  the  variables,         .  218 

410.  Case  of  greatest  simplicity  in  the  de- 

termination of  the  multiplier,     218-219 

411.  Substitution    of  the    arbitrary   con- 

stants for  their  equivalent  functions,  219 

412.  Reduction  of  the  form  when  one  of 

the  A-ariables  is  a  solution,       .  219-220 

413.  Transformation  of  the  multiplier  with 

the  change  of  variables  (21 1„),  220-221 

414.  Corresponding  transformation  of  the 

equations  for  the  determination  of 
the  multiplier  (2210.,),    .         .         .221 

415.  Transformation  when  the  element  of 

differentiation  remains  unchanged 
(261.^,0,        .         .         .         .         .  221 

416.  Transformation  with  a  partial  change 

of  variables  (222^),         .         .         .  222 

41 7.  Transformation    when   the   common 

element  of  differentiation  is  a  va- 
riable,     222 

418.  Transformation   when    part    of    the 

new   variables    are   solutions,  and 
the  multiplier  is  unchanged  (22205),  222 

419.  Transformation  when    pai't   of   the 

new  variables  are  solutions  and  the 
element  is  unchanged,     .         .  222-223 

420.  Transformation   when   all   the   new 

variables  are  solutions  (2283),         .  223 

421.  The  multiplier  of  differential  equa- 

tions of  a  higher  order  expressed  in 
the  normal  form  (222o,-,),         .         .  223 


ANALYTICAL  TABLE  OF  CONTENTS. 


XXV 


422.  Case  in  wliich  the  multiplier  of  dif- 

ferential equations  of  a  higher  or- 
der is  unity  (223a,),        •         •         .  223 

423.  E(|uation  for  the  multiplier  of  differ- 

ential equations  of  a  higher  order 
■when  they  are  not  in  the  normal 
form  (2243i), 224 

424.  Case  in  which  the   equation  for  the 

multiplier     admits     of     reduction 

(2254.  !„)> 225 

425.  Case  in  which  the  given  equations 

cannot  be  reduced  to  the  normal 
form  without  differentiation,  .         .  225 
42G.  Direct  determination  of  the  functions 
involved  in  the  equation  of  the  mul- 
tiplier from   the   given    equations, 

225-226 

427.  Determination  of  the  factor  for  the 

passage  from  the  multiplier  of  the 
given  equation  to  that  of  one  of 
the  simplest  forms  of  normal  equa- 
tions (2288), 226-228 

428-43L  Principle  of  the  Last  Mul- 
tiplier,        ....  228-231 

428.  The   Jacobian    multiplier    coincides 

with  the  Eulerian  multiplier  when 
there  are  two  variables  (229o),  228-229 

429.  Jacobi's  princij^le  of  the  last  mul- 

plier, 229 

430.  By  the  principle  of  the  last  multiplier, 

when  the  element  of  imriation  is  7iot 
directly  expressed  in  the  given  equa- 
tions, either  the  tico  last  integrals 
can  he  obtaitied  by  quadratures,  or 
the  last  integral  can  he  obtained  iviih- 
out  integration,        ....  229 

431.  The  principle  of  the  last  multiplier 

when  a  portion  of  the  variables  are 
not  involved  in  the  remaining  de- 
rivatives, and  the  remaining  de- 
rivatives satisfy  a  given  equation, 

230-231 

432-441.  Partial  Multipliers,    .  231-235 

432.  Definition  of  the  partial  multiijliers 

(231i0, 231 

d 


433,  434.  Defining  equation  of  the  partial 

multipHer  (231o8),  (232j),        .  231-232 

435.  Determination  of  the  signs  in  the  for- 

mation of  the  multipliers  (232,:,),    .  232 

436.  Case  in  which  the  partial  multiplier 

is  the  Jacobian  multiplier,     .         .  232 

437.  Case  in  which  the  partial  multijilier 

is  the  Eulerian  multiplier  amplified 
by  Lagrange,      ....  232 

438.  Every  partial  multiplier  corresponds 

to  an  integral  of  the  equation, .         .  233 

439.  The  deduction   of  an  integral  from 

the  Eulerian  multiplier  (23807),  233-234 

440.  Transformation  of  the  partial  multi- 

plier when  there  is  a  change  of  va- 
riables (239.,r)  ....  239 

441.  Transformation    and     reduction    of 

the  partial  multiplier  -ndien  the 
solutions  are  adojsted  as  new  vari- 
ables,        ....         234,  235 

III.  Integrals  of  the  Differential  Equa- 
tions OF  Motion. 

442.  General  integrals  of  the  equations  of 

motion, 235 

443-451.  The  Application  of  Ja- 
cobi's Principle  of  the  Last 
Multiplier  to  Lagrange's 
Canonical  Forms,        .        236-241 

443.  Lagrange's   canonical   forms  consti- 

tute a  system  of  normal  forms  236 

444.  A    Jacobian     multiplier    is    always 

known  in  equations  of  motion  when 
the  forces  do  not  involve  the  veloc- 
ities (237i8),  .         .    •     .       236-237 

445.  The  principle  of  the  last  multiplier  ex- 

pressed as  a  dynamiccd  principle,    .  237 

446.  The    Jacobian    multiplier    ichen    the 

equations  of  motion  are  expressed 
in  rectangular  coordinants  (23813), 

237-238 

447.  The  conditional  equations  expressed 

in  the  multiplier  of  the  ecpiations 

of  motion  (239,,,),    .         .         .  238-239 

448.  The  transformation  of  the  multiplier 

hy  the   introduction  of  the  original 


XXVI 


ANALYTICAL  TABLE  OF  CONTENTS. 


elements  instead  of  the  rectangular 
coordinates  (44O3),  .  .  .  239-240 
449,  The  Jacobian  multiplier  of  the  equa- 
tions of  motion  when  there  are  no 
equations  of  condition  ;  it  is  unity 
when  the  coordinates  are  rectangu- 
lar (240i,), 240 


450.  The  equations  of  condition  considered 

as  forces  in  the  expression  of  the 
multiplier  (241;),    .         .  .  240-241 

451.  The  multiplier  is  unity  when  the  dif- 

ferential equations  of  motion  are 
expressed  In  Hamilton's  form,    .  241 


CHAPTER    XI. 

MOTION      OF      TRANSLATION, 


452.  The  motion  of  the  centre  of  gravity 

is  Independent  of  the  mutual  con- 
nections, ....  241-242 

453.  The  motion  of  the  centre  of  gravity 

depends  upon  the  external  forces,  242 

454-752.  Motion  of  a  Point,      .  242-433 

454.  The  ditierentlal  equations  of  the  mo- 

tion of  a  point  (2434),     .         .  242-243 

455-459.  A    Point    moving    upon    a 

Fixed  Line,  .         .         .  243-244 

455.  By  the  principle  of  the  multiplier,  the 

motion  is  expressed  hij  integrals  by 
quadratures  (243ig,  24);     •         •         •  243 

456.  The  velocity  dependent  solely  upon 

position,  and  not  upon  the  interme- 
diate path,       ....  243-244 

457.  Case  in  which  the  motion  Is  limited, 

in  which  case  the  oscillations  are 
Invariable  in  duration,    .         .         .  244 

458.  If  the   path  returns  into  itself,  the 

jieriod  of  circuit  is  constant,  .         .  244 

459.  Exjiression   of  the  multiplier  when 

the  forces  and  equations  of  motion 
involve  the  time  (24428),     ■         •       244 

460-477.  The  Motion  OF  a  Body  UPON 
•A  Line  v^^hen  there  is  no  Ex- 
ternal Force.  Centrifugal 
Force,         ....    245-254 

460.  Upon  a  fixed  line  with  no  external 

force  the  velocity  is  constant,  .  245 

461.  Measure    of    the    centrifugal    force 

(245i,), 245 


462.  Total   pressure   upon  a  line   where 

there  are  external  forces,       .         .  245 

463.  The  centrifugal  force  cannot  be  used 

as  a  motive  power,  .         .         .  245 

464.  The  acceleration  of  a  body  upon  a 

moving  line  (24620),        .         .    245-247 

465.  Upon   a  uniformly  moving  line  the 

relative  velocity  of  a  body  acted 
upon  by  no  force  is  constant,  .  247 

466.  The   acceleration  of  a  line   moving 

with  translation  is  a  negative  force 
acting  upon  the  body,    .         .         .247 

467.  The  same  proposition  applies  to  any 

line, 247-248 

468.  Case  in  which  the  line  rotates  uni- 

formly about  a  fixed  axis  (24804),  •  248 

469.  The  time  of  oscillation  of  a  body  up- 

on a  uniformly  rotating  line  Is 
constant,         ....    248-249 

470.  The  period  of  circuit  of  a  body  upon 

a  uniformly  rotating  line  is  constant,  249 

471.  Case  in  which  the  motion  of  the  body 

vanishes  at  the  axis  of  rotation 
(2492,,), 249 

472.  Motion   of  a   body  on   a  uniformly 

rotating  straight  line  (250g2o,  31)5 
(25I2), 249-251 

473.  Motion  of  a  body  on  a  uniformly  ro- 

tating circumference  of  which  the 
plane  Is  perjjendlcular  to  the  axis 
of  rotation  (25I29),  (2523_  22,20)? 
(2533,11,21),      ....    251-253 

474.  Motion   of  a  body  upon   a  rotating 

line  which  is  wholly  contained  up- 


ANALYTICAL  TABLE  OF  CONTENTS. 


XXVU 


on  the  surface  of  a  cylinder  of  revo- 
lution of  which  the  axis  is  the  axis 
of  rotation  (25333),  •         •         .  253    491 

475.  Case  in  which  the  rotation  of  the  cyl- 

inder is  uniform,     ....  254 

476.  Case  in   which  the  curve  is  a  helix 

(254g), 254 

477.  Case   in  which   the   acceleration   is 

uniform  (254i5),      ....  254 


478-482.  MoTiox  OF  A  Heavy  Body 
UPON  A  FIXED  Line.  The  Sim- 
PLK  Pendulum,    .         .         .  254-256 

478.  The  motion  of  a  heavy  body  upon  a 

fixed  line  (25428),  ....  254 

479.  When  the  line  is  contained  upon  the 

surface  of  a  vertical  cylinder,    254-255 

480.  When  the  line  is  straight  (255^),       .  255 

481.  WTien  the  line  is  straight  and  no  ini- 

tial velocity  (2551.,),        .         .         .  255 

482.  When  the  line  is  the  circumference 

of  a  circle ;  oscillations  of  the  sim- 
ple pendulum  (255o5),  (2563,16,530,25), 

255-256 

483-502.  Motion  of    a  Heavy  Body 

UPON  A  moving  Line,  .  257-270 

483.  When  the  line  has  a  motion  of  trans- 

lation (25  7j),  .... 

484.  When  the  translation   is   uniformly 

accelerated,  it  is  equal  to  a  constant 
force,      ...... 

485.  When  the  line   is  straight,  and  the 

law  of  translation  is  given ;  in 
what  case  this  path  Is  a  parabola 
(257^.31),  (258,,  n),  .         .257- 

486.  When  the  translation  of  the  line  is 

uniform  and  direct ;  gain  of  power 
(258,0,  (259i6-25),   •         .         •  258-259 

487.  When  the  line  is  the  circumference 

of  a  vertical  circle  (2593i),  (26019,24), 
(26  lo,  8, 14.18),    ....  259-261 

488.  When  the  line  rotates  about  a  verti- 

cal axis  (26I03),      .         .         .         .261 

489.  When   the    line    rotates    uniformly 

about  a  vertical  axis  (262.),    .  261-262 

490.  When  a   straight  line   rotates  uni- 


257 


257 


-258 


492. 

493. 
494. 

495. 

496. 
497. 
498. 

499. 
500. 

501. 

502. 


263 


264 


formly  about  the  vertical  axis 
(262i3), 262 

Direct  integration  of  the  linear  dif- 
ferential equation  in  this  case  Into 
the  form  given  by  Vieille  (262io),  262 

Case  of  §  490,  In  which  there  is  an 
impassable  limit  (26205,31),  (263„), 

262-263 

Case  of  §  490,  in  which  there  is  no 
hmit  (263ji^), 

Case  of  §  490,  in  which  there  Is  a 
possible  position  of  immobility 
(264i.io),         .... 

When  the  circumference  of  a  circle 
rotates  uniformly  about  a  vertical 
axis ;  the  point  of  maximum  and 
minimum  velocity  defined  by  an 
hyperbola  (265io),  .         .         .  264-265 

Case  of  §  495,  In  which  there  is  no 
motion  upon  the  line,     .        .         .  266 

Case  of-  §  495,  when  the  minimum 
velocity  vanishes  (26 7,,i),         .  266-267 

\^Tien  a  parabola  of  a  vertical  trans- 
verse axis  rotates  uniformly  about 
its  axis  (26810, 31),     .         .         .267-269 

Case  of  §498,  when  the  minimum 
velocity  v^anishes  (269ii), 

When  the  axis  of  rotation  Is  not  ver- 
tical, and  when  the  rotation  is  uni- 
form (269is),  .... 

When  a  straight  line  rotates  imlform- 
ly  about   an  inclined   axis  (270^), 

269-270 

Rotation  of  a  plane  curve  about  an 
Inclined  axis  (270i:j),      .         .         .  270 


269 


269 


503-534.  Motion  of  a  Body  upon  a 
Line  in  opposition  to  Fric- 
tion, OR  through  a  resisting 
Medium,        ....  270-315 


503.  The  resistance  of  a  medium,     . 

504.  Expression  of  the  resistance, 

505.  Resistance  of  a  medium  to  the  mo- 

tion of  a  body  upon  a  fixed  line 
(2713,9),  .         .         .         .         . 

506.  Motion    of  body  upon  a  fixed   line 

through  a  resisting  medium  with- 
out external  force  (271i2,  k),  . 


270 
270 


271 


271 


XXVlll 


ANALYTICAL   TABLE    OF   CONTENTS. 


507. 

508. 
509. 
510. 
511. 
512. 
513. 

514. 
515. 


516- 
51G. 


517, 
518, 


519, 
520, 


521 


Case  of  §  506,  when  the  law  of  resist- 
ance is  expressed  as  a  quadratic 
function  of  the  velocity.  Change 
of  sign  of  the  resistance  (^271o^), 
(2724,13.22),      ....  271-273 

Case  of  §  506,  when  the  resistance 
is  friction  upon  the  line  (273i9),      .  273 

Case  of  §  508,  Avhen  thei"e  is  no  ex- 
ternal force  (273os),        .         .         .273 

Case  of  §  509,  when  the  fixed  line  is 
the  involute  of  the  circle  (274^),    .  274 

Case  of  §  509,  when  the  line  is  the 
logarithmic  spiral  (274i4),       .         .  274 

Case  of  §  509,  when  the  line  is  the 
cycloid  (274._>.,),       .         .  .         .274 

Case  of  §  506,  when  the  resistance  of 
the  line  is  constant,  and  the  resist- 
ing medium  moves  with  a  uniform 
velocity,  and  the  resistance  is  pro- 
portional to  the  velocity  (2753),  274-275 

Case  of  §513,  when  the  line  is 
straight,  and  there  is  no  external 
force  (275„_2o),        .         .         .  275-276 

Motion  of  a  heavy  body  upon  a  fixed 
straight  line,  when  the  resistances 
are  friction,  and  that  of  a  moving 
medium  which  resists  as  the  square 
of  the  velocity  (27631),  (2773,29), 
(278i6),  (2793,15,28),  (280io,2o),     276-280 

-534.  The  Simple  Pendulum    in 

A   RESISTING    MeDIUM,  .    281-315 

The  small  oscillations  of  a  pendulum 
against  friction  and  the  resistance 
of  a  medium  which  is  proportional 
to  the  velocity  (28I20.20),         .         .  281 

The  oscillation  after  many  vibra- 
tions (2824),  .         .         .         •         .282 

The  time  of  oscillation  compared 
with  that  in  a  vacuum  (282ig,23),    .  282 

The  arc  of  oscillation  (283.,),     .    282-283 

The  law  of  diminution  of  the  arc  of 
oscillation  and  of  the  maximum  of 
velocity  (283.,8),      .         .         .         .283 

The  oscillations  of  the  pendulum 
if  the  resistance  is  as  the  square  of 
the  velocity  (28403, 31),     •         •  284-285 


522.  The  arc  of  oscillation  in  the  case  of 

§  521  (2852«) 285 

523.  The  arc  of  oscillation  in  the  case  of 

§  521  is  the  same  as  in  a  vacuum 
(28612,17,0.2),    ....    285-286 

524.  The    oscillations   of    the    pendulum 

when  the  law  of  resistance  is  ex- 
pressed as  a  function  of  the  time 
(287„g),  ....  286-287 

525.  The    oscillations    of   the   pendulum, 

as  affected  by  those  produced  in 
the  medium  (28802,04),  (28924_oo), 
(290.2„),  (2914),        .         .         .  287-291 

526.  The  oscillations  of  the  pendulum  as 

affected  by  the  portion  of  the  me- 
dium which  becomes  part  of  the 
pendulum  (291o8_3i),         •         .  291-292 

527.  Constants  of  the  formulae  of  the  oscil- 

lations of  the  pendulum  in  a  resist- 
ing medium  arranged  for  applica- 
tion to  experiment  (29  2i:_i9), .         .  292 

528.  Approximate  form  for  the  best  exper- 

iments in  which  the  friction  is  in- 
sensible (29224_28),  ....  292 

529.  The  French  system  of  weights  and 

measures  adopted  in  the  examina- 
tion of  experiments,        .         .    292-293 

530.  Discussion    of    Newton's     experi- 

ments upon  the  pendulum  in  air 
(293ig.2o,  26-28),  ....  293-294 

531.  Discussion    of    Dubuat's     experi- 

ments upon  the  pendulum  in  air 
and  water  (2956.7,  iwe),    •         .  294-295 

532.  Discussion      of     Borda's      experi- 

ments upon  the  pendulum  in  air 
(2965^,ii_,3,n.i8),       .  .  .   296-297 

533.  Discussion     of     Bessel's     experi- 

ments upon  the  pendulum  in  air, 

(2987_9,  i3_i5,  20-22,   25-2;),     (299ii_i3,   15.17), 

(2990,^22,25-27),.  •         •  .  298-311 

534.  Discussion  of  Baily's   experiments 

upon  the  pendulum  in  air,       .  311-315 

535-559.  The  Tautochrone,        .  316-327 

535.  Definition  of  the  tautochrone,  .         .315 

536.  The  case  of  the  tangential  force  of 

the  tautochrone  when  it  can  be  ex- 


ANALYTICAL  TABLE  OF  CONTENTS. 


XXIX 


pressed  as  a  function  of  the  arc 
(3179), 316-317 

537.  The  equation  of  the  tautochrone  un- 

der the  action  of  a  fixed  force 
(317ie), 317 

538.  The  tautochrone  which  rotates  uni- 

formly about  a  fixed  axis  when 
there  is  no  external  force  (31 73.5),  .  317 

539.  The  case  of  §  538,  when  it  is  a  plane 

curve  (3184.7),         .         .         •  317-318 

540.  The  cycloid  is  the  tautochrone  of  a 

free  heavy  body  in  a  vacuum 
(31823) 318-319 

541.  The  tautochrone  of  a  heavy  body  in 

a  vacuum  upon  a  given  surface 
(319i3), 319 

542.  The  tautochrone  of  §541,  when  the 

surface  is  a  cylinder  of  which  the 
axis  is  horizontal,  and  the  equa- 
tion of  the  base  is  (31905),  (320;), 

319-320 

543.  The  tautochrone  of  §  542  upon  the 

developed  cylinder  (320,4),    .         .  320 

544.  The   tautochrone  of  §542,   when  it 

passes  through  the  lowest  side  of 
the  cylinder  (320i4.,„,.>4),  •  320-321 

545.  The  differential  equation  of  the  tau- 

tochrone of  §542  referred  to  rec- 
tangular coordinates  (32I10.15),        .  321 

546.  The  tautochrone  of  §  542,  when  the 

base  of  the  cylinder  is  a  cycloid 
(32I2.,),  (322^),        .         .  .  321-322 

547.  The  tautochrone  of  a  heavy  body 

upon  a  surface  of  revolution  of 
which  the  axis  is  vertical,  and  the 
meridian  curve  is  that  of  (31 905), 
(322,,), 322 

548.  The   tautochrone   of  a  heavy  body 

upon  a  vertical  cone  of  revolution 
(322^8),  (323,),         .         .         .  322-323 

549.  The   tautochrone    of    §  548,    which 

passes  through  the  vertex  (3239),   .  323 

550.  The   tautochrone    of    §  547,    when 

the  meridian  curve  is  a  cycloid 
(323,0, 323 

551.  The  tautochrone  upon  a  plane  when 

the  force  is  directed  towards  a 
point   in    the   plane,    and  propor- 


tional to  some  power  of  the  dis- 
tance from  the  point  (323,.-,),  .  323 

552.  The  tautochrone  of  §  551,  when  the 

force  is  any  function  of  the  dis- 
tance (324,,;),  ....  323-324 

553.  The  polar  differential  equation  of  the 

tautochrone  in  the  case  of  §  552 
(324^0, 324 

554.  The  difl[erential  er|uation  of  the  tau- 

tochrone of  §  552  in  terms  of  the 
radius  of  curvature  and  the  angle 
of  direction  (324o7),        •         •         .  324 

555.  The  tautochrone  of  §552,  when  it  is 

the  involute  of  the  circle  (325,,),    .  325 

556.  The   tautochrone    of  §552,  when  it 

is    a    logarithmic     spiral    (32526), 

325-326 

557.  The  tautochrone  of  §552,  when  the 

force  is  proportional  to  the  dis- 
tance from  the  origin ;  when  it  is 
not  infinite,  it  is  an  epicycloid 
(32G2,),  (327,o),       .         .         .  326-327 

558.  Cases    included   in  §  557,  near   the 

point  of  greatest  velocity,       .         .327 

559.  The  tautochrone  in   a  resisting  me- 

dium postponed  to  case  of  holo- 
chrone, 327 

560-604.  The  Brachistochroxe,    328-354 

560.  Definition  of  the  brachistochrone,     .  328 

561.  The  investigation  of  the  free  brachis- 

tochrone (328,9),    •         •         •         .328 

562.  The  brachistochrone  when  the  act- 

ing forces  are  fixed  (328,4,  js),         •  328 

563.  The    pressure   upon   the   brachisto- 

chrone is  double  the  centrifugal 
force  (3294),   ....  328-329 

564.  The  point  of  contrary  flexure  in  a 

brachistochrone,     .         .         .         .329 

565.  The    conditions    of    the    brachisto- 

chrone introduced  by  the  general 
method  of  variations,      .         .         .  329 

566.  When  the  force  is  directed  towards 

a  point,  the  free  brachistochrone 
is  a  plane  curve,  and  its  plane  in- 
cludes the  point  of  attraction,         .  329 

567.  When   the   forces   are    j)ai'allcl,  the 

free    brachistochrone   is    a    j)lane 


XXX 


ANALYTICAL  TABLE  OF  CONTENTS. 


curve,  and  its  plane  is  parallel  to 
the  direction  of  the  forces,     .         .329 

568.  When  there  are  no  forces  the  brachis- 

tochrone  is  the  shortest  line,   .  329-330 

569.  The  equation  of  the  brachistochrone 

Avhen  its  force  is  central  (330^-,),    .  330 

570.  The  brachistochrone  of  §  569,  when 

the  force  is  proportional  to  the  dis- 
tance ;  it  is  a  spiral  or  an  epicy- 
cloid (331io,  5„),       .         .         .    330-331 

571.  The  equation  of  the  brachistochrone 

when  the  forces  are  parallel  (33l3j), 

331-332 

572.  The  brachistochrone  of  a  heavy  body 

is  a  cycloid  (332o),  .         .         .332 

573.  The  brachistochrone  of  §  571,  when 

the  force  is  proportional  to  the  dis- 
tance from  a  given  line  (332io), 
(333io),  ....    332-333 

574.  The  centrifugal  force  in  the  brachis- 

tochrone upon  a  given  surface, 

575.  Simple   case    of   a    brachistochrone 

upon  a  given  surface,  including 
that  of  a  meridian  line  upon  a  sur- 
face of  revolution,  .  .    333- 

576.  The  brachistochrone  upon  a  surface 

of  revolution  when  the  force  is  di- 
rected to  a  point  of  the  axis,  and 
expression  of  the  projection  of  the 
area  upon  the  plane  perpendicular 
to  the  axis  (334,g_.,o),       .         .    334-335 

577.  The  derivatives  of  the  arc  and  of  the 

difference  of  longitude  in  the  case 
of  §  576,  taken  with  reference  to 
the  arc  of  the  meridian  (3355.7), 

578.  The  derivatives  of  the  same  quanti- 

ties taken  with  reference  to  the 
latitude  (331ii_i,j),  .... 

579.  The  surface  upon  which  the  brachis- 

tochrone may  make  a  constant 
angle  with  the  meridian  ;  it  may 
be  used  to  define  the  limits  of  the 
brachistochrone  in  any  case  of 
§  576  (335,„),  (336o),      .         .    335-336 

580.  The  limiting    surface    of  §  579  is  a 

paraboloid  of  revolution  in  the  case 
of  a  heavy  body,  of  which  the  axis 
is  directed  downwards.     Investiga- 


333 


-334 


335 


335 


581 


582. 


583, 


584. 


586. 


587. 


589. 


590. 


591. 


tion  of  the  other  brachistochrones 
upon  this  surface  (33629),  (33  73  j^^  „;), 

(3382.  7^  i3_  ir|_oo),  (3395,  13^  oe))  (3407_  14), 

336-340 

The  brachistochrone  for  a  heavy 
body  upon  a  paraboloid  of  revolu- 
tion of  which  the  axis  is  the  upwai'd 
vertical  (34O21),  (341s),  •         •  340-341 

The  brachistochrone  of  a  heavy  body 
upon  a  vertical  right  cone  (341 20), 
(3422.5, 10. 23),  (3432, 13, 10, 22. 25),  (3442), 

341-344 

The  brachistochrone  of  a  heavy  body 
upon  an  ellipsoid  of  revolution  of 
which  the  axis  is  vertical  (344i4), 
(345io, 05),  (3460.6,  j„9, 25),.         .  344-346 

The  tangential  radius  of  curvature  of 
the  brachistochrone  of  a  heavy  body 
upon  any  surface  (3477),         .    346-347 

When  the  force  is  parallel  to  the  axis, 
and  proportional  to  the  distance 
from  a  plane  which  is  perpendicular 
to  the  axis,  the  limiting  surface  of 
§  579  is  an  ellipsoid  or  an  hyper- 
boloid,    ..... 

When  the  force  is  proportional  to  the 
distance  in  §  576,  the  limiting  sur- 
face of  §  579  is  an  ellipsoid  or  an 
hyperboloid, 

Investigation  of  the  limiting  surface 
of  §  579  when  the  force  is  propor- 
tional to  the  square  of  the  distance 
in  §  576,  .... 

The  normal  pressure  upon  the  brach- 
istochrone when  the  length  of  the 
arc  is  given  (347._h,),         .         .347- 

The  equation  of  the  brachistochrone 
in  the  case  of  §  569,  when  the  length 
of  the  arc  is  given  (348,5,10),   . 

The  equation  of  the  brachistochrone 
in  the  case  of  parallel  forces  when 
the  length  of  the  arc  is  given 
(348h,:s),         .... 

The  equation  of  the  brachistochrone 
in  the  case  of  §  576,  when  the 
length  of  the  arc  is  given.  The 
investigation  of  the  limiting  sur- 
face (34805,09),  (3495),     •         •    348-349 


347 


347 


347 


-348 


348 


348 


ANALYTICAL  TABLE  OF  CONTENTS. 


XXXI 


349 


349 


592.  The  normal  pressure  in  a  brachisto- 

chrone  when  the  total  expenditure 
of  action  is  given  (349io), 

593.  The  equation  of  the  brachistoohrone 

in  the  case  of  §  576,  when  the  to- 
tal expenditure  of  action  is  given 

(34924,27), 

594.  The  equation  of  the  brachistochrone 

in  the  case  of  parallel  forces  -when 
the  total  expenditure  of  action  is 
given  (34931),  (350.,),      .         .  349-350 

595.  The  equation  of  the  brachistochrone 

in  the  case  of  §  576,  when  the  to- 
tal expenditure  of  action  is  given, 
and  the  investigation  of  the  limit- 
ing surface  (350s_  10,19),  .         .         .  350 

596.  The  brachistochrone  in  a  medium  of 

constant  resistance  (35l3_g,  13),    350-351 
697.  The  expression  of  the  multiplier  of 
the    equation  for   the    element   of 
length  of  the  arc  when  the  force  is 
central  in  the  case  of  §  596  (35I20),  351 

598.  The    expression    of    this    multiplier 

when  the  forces  are  parallel  (35I24),  351 

599.  The  ecpiation  of  the  brachistochrone 

of  a  heavy  body  in  a  medium  of 
constant  resistance  (35I29),     .         .  351 

The  brachistochrone  in  a  medium, 
of  which  the  resistance  is  a  given 
function  of  the  velocity  (352]o^ig  05),  352 

The  equations  of  §  600  when  the 
forces  are  parallel  (35209^1),  .         .  352 

602.  The  brachistochrone  of  a  heavy  body 

in  any  resisting  medium,  and  in  the 
case  of  the  resist<ince  inversely  pro- 
portional to  the  velocity,  and  di- 
rectl}-  propoi'tional  to  the  square  of 
the  velocity,  (3533_g,  20, 24-2o)»     • 

603.  Euler's  error  in  regard  to  the  nor- 

mal pressure  of  a  brachistochrone 
in  a  resisting  medium,    . 

604.  Singular  difficulty  in  the  special  de- 

termination of  the  brachistochrone 
when  its  form  is  given.  Special 
example  of  such  an  investigation 
(354i3), 


600. 


601. 


353 


353 


605-624.  The  Holochroxe, 


354 


354-364 


605. 
606, 


607. 
608, 

609. 
610. 
611, 
612, 

613, 
614, 
615. 
616. 


617. 


618. 


Definition  of  the  holochrone,     . 

The  force  along  the  curve  of  the 
holochrone  when  the  forces  are 
fixed  (3554,9),         .         .         .    354- 

The  holochrone  for  a  heavy  body 
(355i3), 

The  force  along  the  holochrone  when 
the  time  of  descent  admits  of  de- 
velopment according  to  integral 
ascending  powers  of  the  arc 
(.35oig_3i),         ..... 

Given  function  of  the  initial  value  of 
the  potential  (3579,15),    .         .    356- 

Case  of  §  609  when  the  forces  are 
parallel  (3572.,),      .... 

Case  of  §  609  when  the  forces  are 
central  (357.,;,),       .         .         .         . 

Case  of  §  609  when  the  time  is  devel- 
oped according  to  powers  of  the 
initial  value  of  the  potential  (3585), 

Case  of  §  609  when  the  curve  of  ap- 
proach to  the  point  of  maximum 
potential  is  given,  and  the  whole 
time  is  a  given  function  of  the 
maximum  potential  (358ij),    . 

Case  of  §  609  when  the  time  of  os- 
cillation is  constant,  which  is  a  cu- 
rious sjieeies  of  tautochrone  inves- 
tigated byEuLER  for  heavy  bodies 
(358.2,),    ' 

The  holochrone  is  indeterminate 
when  the  forces  may  depend  upon 
the  velocity,  but  there  is  a  con- 
dition which  must  be  satisfied 
(359i„),  ....    358- 

The  case  of  §  615  when  the  force 
along  the  curve  may  be  separated 
into  two  parts,  of  which  one  is  fixed 
and  depends  upon  the  arc,  while 
the  other  depends  upon  the  veloc- 
ity, lias  been  largely  discussed  with 
little  success  and  much  animosity  . 

The  case  of%  615  transformed  to  L.\- 
grange's  most  general  form  of  the 
tautochrone  (35928), 

The  case  of  §  615  transformed  to  La- 
place's general  form  of  the  tauto- 
chrone (360^),        .... 


354 

355 
355 

355 
357 
35  7 
357 

358 
358 
358 
359 


359 


359 


360 


xxxu 


ANALYTICAL  TABLE  OF  CONTENTS. 


360 


3G0 


619.  The   case  of  §  615  transformed  to  a 

third  form  equally  general  with 
those  of  Lagrange  and  Laplace 
(360„), 

620.  The  case  of  §  615  when  the  assumed 

equation  consists  of  three  parts, 
which  are  functions  respectively 
of  the  are,  the  time,  and  the  veloc- 
ity (360,,),     

621.  Transformatio  n  of  the  preceding  form 

to  a  familiar  formula  of  La- 
grange (36 1.),      .         .         .  360-361 

622.  Case  in  which  the  form  of  §621  co- 

incides with  that  of  §616,  which 
sustains  the  correctness  of  Fon- 
taine's strictures ;  and  this  holo- 
chrone  is  essentially  tautochronous 
(362,), 361-362 

623.  Case  of  §  615,    which  includes   La- 

grange's formula  (3623i),  (363,,), 

362-363 

624.  Case  of  §621,   in   which   the   force 

along  the  curve  has  the  form  given 
in  §616,  with  the  addition  of  a 
term  which  is  the  product  of  the 
square  of  the  velocity  by  a  function 
of  the  arc  (3 6 3,1),  .         .         .  363-364 

625-639.  The  Tachytrope,  .         .  364-368 

625.  Definition  of  the  tachytrope,     .         .  364 

626.  Application  of  §615  to  the  tachytrope,  364 

627.  Case  in   which  the  time   is   not   in- 

volve 1  in  the  assumed  equation  of 
§  615,  and  the  force  has  the  form 
of  §616  (364io),      ....  364 

628.  Case  of  a  heavy  body  in  which  the 

tachytrope  is  a  cycloid  (364.3,3,), 
(365,), 364-365 

629.  Klingstieuna's  case  of  the  tachy- 

trope in  a  medium  which  resists  as 
the  square  of  the  velocity,  which 
was  solved  by  Clairaut  (365oi),    365 

630.  Case  of  §  627,  when  the  velocity  is 

uniform  (SeS^j),     ....  365 

631.  Case    of  the   tachytrope    when    the 

forces  are  parallel,  and  the  as- 
sumed equation  of  §615  does  not 
involve  the  time  (366,),         .         .  306 


632.  Case   of  §631,   when    the   velocity 

has  a  constant  ratio  to  that  in  a 
vacuum  (36612,19),.         .         .         .  366 

633.  Case  of  §627,  when  the  forces  are 

central,  and  the  assumed  equation 
is  expressed  in  terms  of  the  veloc- 
ity and  the  radius  vector  (36634),   •  366 

634.  Case  of  §633,  when  the  velocity  has 

a  constant  ratio  to  that  in  a  vacuum 
(36631),  (367,),         .         .         .  366-367 

635.  Case  in  which  the  velocity  in  a  given 

direction  is  a  given  function  of  the 
arc  and  the  distance  in  that  direc- 
tion (36  7i.,), 367 

636.  Case  in  which  the  velocity  in  a  given 

direction  is  uniform  (36 7i,j),    .         .36  7 

637.  Case   of  §    636    for   a   heavy    body 

(367,5, 31), 36  7 

638.  Case  of  §  637  when  there  is  no  re- 

sisting medium  ;  for  a  horizontal 
direction  the  tachytrope  is  a  para- 
bola, and  for  a  vertical  direction 
it  is  the  evolute  of  a  parabola 
(3683,8,15,0,), 368 

639.  The  tachytrope  of  a  heavy  body  when 

the  resistance  is  proportional  to  the 
velocity  (36831),      .         .         .         .368 

640-646.  The  Tachistotrope,       .  369-370 

640.  Definition  of  the  tachistotrope,  .  369 

641.  The   tachistotrope   in   a  medium  of 

which  the  resistance  is  a  given 
function  of  the  velocity  (369i4_2i),  .  369 

642.  The  normal  pressure  on  the  tachis- 

totrope when  the  resistance  is  pro- 
portional to  a  power  of  the  veloc- 
ity (369,,), 369 

643.  The   tachistotrope  is  a  straight  line 

when  the  resistance  is  constant,      .  370 

644.  The  tachistotrope  for  parallel  forces 

(37O5), 370 

645.  The  tacliistoti'ope  of  a  heavy  body 

(370i,„o), 370 

646.  The   tachistotrope  of  a  heavy  body 

when    the    resistance    is    that    of 

§  642  (370i„),  .  .         .  .370 

64  7-655.  The    Barytrope    and    the 

Tautobrayd,       .         .         .  370-373 


ANALYTICAL  TABLE  OF  CONTENTS. 


XXXlll 


647.  Definition  of  the  barytrope  andtauto- 

barvd, 370 

648.  Tlie  barytrope   when  the  force  has 

the  form  of  §  GIG  (37 O31),        .         .370 

649.  The  barytrope  and  tautobaryd  of  a 

heavy  body  (371,59),      .         .         .371 

650.  The  barytrope  and  tautobaryd  when 

the  resistance  is  cpnstant ;  this  in- 
vestigation is  applied  to  a  heavy 
body  (371,,,i,,,„,3,),  (3723),      .  371-372 

651.  The  barytrope  against  which  there 

is  no  pressure  is  the  curve  of  free 
motion  (372io),        .         .         .         .372 

652.  When  the  curve  of  the  barytrope  is 

given,  the  relations  of  the  fixed 
force  and  resistance,       .         .         .372 

653.  These  relations  in  the  case  of  parallel 

forces  (37219,2^),      .         .         .         .372 

654.  The  relations  of  §  653  applied  to  the 

circle  (37209),  (3783),      .         .  372-3  73 

655.  The  relations  of  §G53  applied  to  a 

cycloid  (373i._,  10),    .         .         ,         .373 

656-662.  The  Synchrone,     .         .  373-374 

656.  Definition  of  the  synchrone  and  its 

dynamic  pole,         .         .         .         .373 

657.  The  synchrone  for  a  constant  time, 

373-374 

658.  The  synchrone  in  a  resisting  medium 

without  force,  on  a  path  of  given 
form  is  the  surface  of  a  sphere,       .  374 

659.  The  synchrone  for  a  uniformly  ro- 

tating straight  line  without  exter- 
nal force  is  a  surface  of  revolution,  3  74 

660.  The    synchrone    for    certain    fixed 

forces  upon  straight  lines  is  a  sur- 
face of  revolution, .         .         .         .374 

661.  The  synchrone  of  a  heavy  body  with- 

out resistance  (3  742o),    .         .         .  374 

662.  The  synchrone  of  a  heavy  body  in  a 

medium  which  resists  as  the  square 

of  the  velocity  (37431),    .         .         .374 

663-670.  The  Syntachyd,   .        .    375-376 

663.  Definition  of  the  syntachyd,     .         .375 

664.  Investigation  of  the  syntachyd,      .       375 

665.  In  the  case  of  §  658,  the   syntachyd 

coincides  with  the  synchrone,         .  3  75 


666.  In  the  cases  of  §§659   and   660,  the 

syntachyd  is  a  surface  of  revolu- 
tion,        375 

667.  When   the    action    is   that   of  fixed 

forces,  the  syntachyd  is  a  level  sur- 
face,         375 

668.  The   syntachyd   for   a    heavy   body 

moving  upon  a  straight  line  against 
a  constant  friction  and  through  a 
niedimn  of  which  the  resistance  is 
proportional  to  the  square  of  the 
velocity  (3  75.,4),     .         .         .         .375 

669.  The  syntachyd  in  a  case  like  that  of 

§  668,  but  in  which  the  resistance  of 
the  medium  is  proportional  to  the 
velocity  (3 76o),.      .         .         .375-376 

670.  The   syntachyd   for   any  body  with 

the  resistances  of  §  668  (3  76,,),        .  376 

671-735.   A    Point    moving    upon    a 

Fixed  Surface,  .        .        .  376-423 

671.  The  motion  of  a  point  upon  a  fixed 

surface  (377„),         .         .         .  376-377 

672.  The    centrifugal    force   of    a    body 

against  a  surface  (3 7 717),        .         .377 

673.  When  the  force  is  normal  to  the  sur- 

face, the  path  is  the  shortest  line,  .  377 
6  74.  When  the  velocity  is  constant,  the 
body  moves  upon  the  intersection 
of  the  given  surface  with  a  level 
surface,  .         .         .         .         .         .377 

675.  When  the  velocity  is  a  given  func- 

tion of  the  parameter  of  the  level 
surface,  the  equation  of  a  second 
surface  upon  which  the  body 
moves,    .         .         .         .         .         .377 

676.  When  the  force  is  directed  toward 

the  origin,  the  area  described  by 
the  radius  vector  is  proportional 
to  the  time.  The  polar  ec][uation 
of  the  path  (37820),         .        .         .378 

677.  When  the  force  of  §  676  is  attractive 

and  inversely  proportional  to  a 
power  of  the  distance,  and  the  ve- 
locity is  that  obtained  by  falling 
from  an  infinite  distance,  the  polar 
equation  of  the  path  admits  of  sim- 
ple integration ;  in  gravitation  the 


XXXIV 


ANALYTICAL  TABLE  OF  CONTENTS. 


path  is  a  parabola  ;  for  a  force  in- 
versely proportional  to  the  cube 
of  the  distance,  it  is  a  logarithmic 
spiral;  for  a  force  inversely  pro- 
portional to  the  fourth  power  of 
the  distance,  it  is  an  epicycloid ; 
for  a  force  inversely  proportional 
to  the  fifth  power  of  the  distance,  it 
is  the  circumference  of  a  circle ;  for 
a  force  inversely  proportional  to 
the  sixth  power  of  the  distance,  it  is 
the  trifolia;  for  a  force  inversely 
proportional  to  the  seventh  power 
of  the  distance,  it  is  the  lemniscate ; 
for  a  repulsive  force  proportional 
to  the  distance,  it  is  an  equilateral 
hyperbola  (379=,),  .         .         .    379-380 

678.  Case   in    which    the    integration   of 

§  676  is  simple  (380.„),  .         .         .380 

679.  Case  of  §  6  78  for  a  force  of  four  terms 

one  of  which  is  constant,  and  the 
others  are  respectively  proportional 
to  the  distance  and  to  its  inverse 
square  and  cube  (381;,  n,  20,  a:-,  si)  > 
(3823,6), 380-382 

680.  Case   of  §  678   for  a  force   of  four 

terms  which  are  inversely  i)ropor- 
tional  to  the  second,  third,  fourth, 
and  fifth   powers   of    the  distance 

(382i5,  19,  24,  27,  3l))         ....    382 

681.  Case  of  §  678  for  a  binomial  form  of 

the  radical  of  (37800),  (3885, «),       .  383 

682.  The  general  forms  of  force  of  §  676 

which  admit  of  simple  integration 
consist  of  two  terms,  of  which  one 
is  inversely  proportional  to  the 
cube  of  the  distance,  and  the  other 
is  proportional  to  the  distance  or 
inversely  proportional  to  the  square 
of  the  distance  (383i3),  .         .         .383 

683.  The  term,  wJiich  is  inversely  propor- 

tional to  the  cube  of  the  distance,  does 
not  increase  the  difficulty  of  integra- 
tion, and  the  effect  of  this  term  may 
be  disguised  in  the  constants  (3833,), 

383-384 

684.  Case  of  no  force  and  of  a  central 

force  inversely  proportional  to  the 


cube  of  the  distance  (384,o,2i,27)» 
(3853, 6,  K,),      ....    384-385 

685.  Case  of  a  central  force  proportional 

to  the  distance,       .         .         .    385-386 

686.  Case  of  §  685  combined  with  a  force 

inversely  proportional  to  the  cube 

of  the  distance  (38 6i5_ig),        .         .386 

687.  Case  of  a  central  force  inversely  jjro- 

jjortional  to  the  square  of  the  dis- 
tance,          386-387 

688.  Case  of  §  687  combined  with  that  of 

§  684  (387,^28),  (388i„_„),        .    387-388 

689.  Case   of   §   683    Avith   the   force   of 

§  684  (388og),  .         .         .         .388 

690.  The  general  laws  of  force  for  which 

integration  may  be  effected  by  el- 
liptic integrals,  each  consisting  of 
four  terms,  with  a  total  variety  of 
six  cases  (3884,8),  .         •         •    388-389 

691.  Second  case  of  the  first  form  of  §  690 

when  the  force  consists  of  terms 
of  the  form  of  §  679  (390o„^,), 
(391i2-„),  (392o,),(393«,  19-23),  (3  94^„), 
(394,9.04),  (396,0,  (397,.;),  (3972,.24), 
(3983,2(^23),  (399,_9,i2),      .         .  389-399 

692.  First  case  of  the  first  form  of  §  690, 

when  the  four  terms  are  respec- 
tively projiortional  to  the  distance, 
to  its  third  and  fifth  powers,  and 
to  its  inverse  cube  (399i5, 19, 22-31), 
(400,_,8),  ....  399-400 

693.  Case  of  §  692,  in  which  the  force  is 

proportional  to  the  fifth  power  of 
the  distance, 400 

694.  Case  of  §692,  in  which  the  force  is 
,     proportional  to  the  cube  of  the  dis- 
tance,      401 

695.  Third  case  of  the  first  form  of  §  690, 

in  which  the  four  terms  are  in- 
versely proportional  to  the  cube 
root  of  the  distance,  to  the  fifth  and 
seventh  powers  of  the  cube  root, 
and  to  the  cube  of  the  distance 
(401,5,  ,8, 25), 401 

696.  Case  of  §  695,  in  which  the  force  is 

inversely  proportional  to  the  cube 
root  of  the  distance,         .         .  401-402 

697.  Case  of  §695,  in  which  the  force  is 


ANALYTICAL  TABLE  OF  CONTENTS. 


XXXV 


inversely  proportional  to  the  fifth 
and  the  seventh  powers  of  the  cube 
root  of  the  distance,        .         .         .  402 

698.  Fourth  case  of  the  first  form  of  §  690, 

in  which  the  four  terms  are  inverse- 
ly proportional  to  the  square  and 
cube  of  the  distance,  and  the  third 
and  fifth  powers  of  the  square  root 
(402.,;. 09),  (403,),    .         .         .    402-403 

699.  Cases  of  §698,  in  which  the  force  is 

inversely  proportional  to  the  third 
and  fifth  powers  of  the  square  root 
of  the  distance,       ....  403 

700.  First  case  of  the  second  form  of  §  690, 

in  which  the  four  terms  are  inverse- 
ly proportional  to  the  second,  third, 
fourth,  and  fifth  powers  of  the  dis- 
tance (403.,,),  (4043,5),    .         .  403-404 

Case  of  §  700,  in  which  the  force  is 
inversely  proportional  to  the  fourth 
power  of  the  distance,    .         .         .  404 

Case  of  §  700,  in  which  the  force  is 
inversely  proportional  to  the  fifth 
power  of  the  distance,     .         .  404-405 

Second  case  of  the  second  form  of 
§  690,  in  which  the  four  terms  are 
proportional  to  the  distance,  and 
inversely  proportional  to  the  third, 
fifth,  and  seventh  powers  of  the 
distance  (40014,16,03),       .         .         .  405 

704.  Case  of  §  703,  in  which  the  force  is 

inversely  proportional  to  the  sev- 
enth power  of  the  distance,     .  405-406 

705.  Third  form  of  central  tbrce,  in  which 

the  integration  can  be  performed 
by  elliptic  integrals  (406^,27,31)5      •  406 

706.  The  potential  curve  for  defining  the 

limits  of  the  path  described  under 
the  action  of  a  central  force  (407io),  407 

707.  The  term  of  the  jjotential,  which  cor- 

responds to  the  force  of  §  686,  may 
be  omitted  in  the  potential  curve 
of  §706, 407 

708.  Potential  curve  in  which  the  path  can 

only  consist  of  a  single  portion,  407—408 

709.  The  portion  of  the  potential  curve 

which  corresponds  to  attraction  and 
repulsion,        .....  408 


701 


702 


703 


710. 

711. 

712. 

713. 
714. 
715. 

716. 
717. 

718. 

719. 

720. 
721. 


723 


724 


Form  of  the  path  for  a  central  force 
in  the  vicinity  of  the  centre  of  ac- 
tion,           408-409 

Character  of  the  path  for  a  central 
force  at  an  infinite  distance  from 
the  centre  of  action,       .         .         .  409 

Graphic  determination  of  the  incli- 
nation of  the  path  to  the  radius 
vector, 409 

The  equation  of  the  path  for  parallel 
forces  (4IO9),  .         .         .         .410 

The  path  of  a  projectile  is  a  para- 
bola (41O18), 410 

The  equation  of  the  curve  for  par- 
allel forces  referred  to  rectangular 
coordinates  (41O31),         .         .         .410 

The  potential  curve  for  parallel 
forces, 411 

Case  in  which  the  force  of  §  713  is 
proportional  to  the  distance  from 
a  fixed  line  (411  i2_i5),     .         .         .  411 

Case  in  which  the  force  of  §  713  is 
proportional  to  the  distance  from 
any  fixed  line  divided  by  the 
square  of  the  distance  from  an- 
other line  (41L,,;),  .         .         .         .411 

The  motion  of  a  body  upon  a  surface 
of  revolution  when  the  force  is  cen- 
tral, and  the  centre  of  action  is 
upon  the  axis  of  revolution  (4123^), 

411-412 

Derivatives  of  the  arc  and  of  the 
longitude  in  the  case  of  §719 
(412„.3,i5.n), 412 

Case  of  §719,  in  which  the  path  of 
the  body  makes  a  constant  angle 
with  the  meridian.  The  surface  of 
revolution  which  defines  the  limits 
ofthe  path  (41 2oi),  (4135),     .   412-413 

Limiting  surface  of  revolution  for  a 
heavy  body  (413,0,        •         •         -413 

Motion  of  a  heavy  body  upon  a  verti- 
cal right  cone  (4 1 43_ii,  22-31) ,  (4 1 5 j_n) , 
(415is.25,3i),  (4I63),.         .         .413-416 

Motion  of  a  heavy  body  upon  a  verti- 
cal paraboloid  of  revolution  of  which 
the  axis  is  directed  downwards 
(416i,_.,3,  ,^.31),  (4172,11.1,0,         •    416-417 


XXXVl 


ANALYTICAL  TABLE  OF  CONTENTS. 


725.  Motion  of  a  heavy  body  upon  a  ver- 

tical paraboloid  of  revolution  of 
■vvhicli  the  axis  is  directed  upwards 
(4172.^), 417 

726-735.  The  Spherical  Pendulum, 

418-423 

726.  The  path  of  a  spherical  penduliun. 

(418,8,  ,0, 418 

727.  Relation  of  the  limits  of  the  path  of 

the  pendulum  (41S,-^.,),  .         .418 

728.  The  time  of  oscillation  for  different 

lengths  of  pendulums,     .         .   418-419 

729.  Case  in  which  the  path  of  the  pendu- 

lum is  a  horizontal  circle  (41  On,  17,23)) 
(4208, „),  .         .         .         .  419-420 

730.  The  time  of  a  complete  revolution  in 

the  case  of  §  729   (420^4),   (421^, 

420-421 

731.  The  path   of  the    pendulum  when 

it  is  nearly  a  horizontal  circle 
(421,0.18),     '   •         •         •         •    .    .•  421 

732.  The  path  of  the  pendulum  when  it  is 

nearly  a  great  circle  (42109,31),  421-422 

733.  The  path  of  the  pendulum  when  it 

passes  nearly  through  the  lowest 
point  of  the  sphere  (422io,  12),  .  422 

734.  Limits  of  the  arc  of  vibration  of  the 

pendulum,      .....  422 

735.  The  azimuth  of  the  pendulum  (4232o,  25), 

422-423 

736-752.  The    Motion     of    a    Free 

Point, 424-433 

736.  The  acceleration  of  a  free  point  in 

any  direction  (4246),      •         •         .424 

737.  The  rotation-area  with  reference  to 

the  moment  of  the  force  about  an 
axis  (4242g),    ....    424-425 

738.  The  potential  for  a  central  force  pro- 


425 


425 


portional  to  the  distance.  The 
path  is  a  conic  section  (425i2), 

739.  The  area  for  forces  directed  towards 

a  line  is  proportional  to  the  time, 

740.  The  path  for  the  case  of  forces  di- 

rected towards  a  line  investigated 
by  means  of  the  peculiar  coordi- 
nates of  the  distances  from  two 
fixed  points  of  the  line  (426i,,  31), 
(4273),  (428i„.,9),    .         .         .    425-428 

741.  The  special  cases  of  §740  may  be 

combined  into  one  by  addition,      .  428 

742.  Cases  in  which  the  forms  of  §  740 

are  expressed  by  elliptic  integrals 
(42807.09),         ....    428-429 

743.  Case  of  §  740,  in  which  there  are  two 

forces  which  follow  the  law  of 
gravitation,    .... 

744.  Case    of  §  740,    in   which   there   is 

one  force  proportional  to  the  dis- 
tance,    ..... 

745.  Case  of  §  740,  in  which  there  is  one 

force  inversely  proportional  to  the 
distance  from  the  fixed  line,  .   429-430 

746.  Restriction  of  the  law  of  force  for 

motion  upon  a  given  curve,   . 

747.  Bonnet's  theorem  for  combination 

of  forces  which  produce  a  given 
motion  (43O30),        .... 

748.  General  value  of  the   potential  for 

§  746  (431e), 

749.  Cases  in  which  the  curve  of  §  746  is 

a  parabola  (431,7.i9,oo_,4,3o^ii),  . 

750.  Case  in  which  the  curve  of  §  746  is  a 

conic  section  (432;.io),    . 

751.  Case  in  which  the  curve  of  §  746  is  a 

cycloid  (432o3_oj),    . 

752.  Case  in  which  the  curve  of  §  746  is  a 

circle,  or  in  which  the  surface  of 
free  motion  is  a  sphere,  .         .  432-438 


429 


429 


430 


430 


431 


431 


432 


432 


CHAPTER    XII. 

MOTION      OF      rotation. 

753.  Rotation-area  defined.     Principle  of         i  754.    The     parallelopiped     of     rotation- 

the  conservation  of  areas,       .   433-434  1  areas, 434 


ANALYTICAL  TABLE  OF  CONTENTS. 


XXXVU 


755. 
756. 


758. 
759. 


760. 
161, 


797.  Rotation  of  a  Solid  Body, 

434-458 

The  moments  of  inertia  and  the  in- 
verse ellipsoid  of  inertia,         .   434-436 

Rotation  about  a  principal  axis  pro- 
duces no  rotation-area  about  the 
other  principal  axes,       .         .         .  436 

The  plane  of  maximum  rotation-area 
is  conjugate  to  the  axis  of  rotation,  436 

The  ellipsoid  of  inertia,    .         .         .  436 

Position  of  the  axis  of  maximum  ro- 
tation-area with  reference  to  the 
axis  of  rotation  in  the  direct  and 
inverse  ellipsoids  of  inertia,     .  436-437 

Ecler's  equations  for  the  rotation  of 
a  solid  (43  7oo,3o),    ....  437 

The  equation  of  living  forces  in  the 
rotation  of  a  solid  (4383, 9),      .   437-438 


762. 


763. 


762-769.  Rotation  of  a  Solid  Body 
which  is  srb.ject  to  no  ex- 
TERNAL Action,  .         .         .    438-443 

The  velocity  of  rotation  of  this  solid 
is  proportional  to  the  correspond- 
ing diameter  of  the  inverse  ellip- 
soid (4380,,), 438 

The  velocity  of  rotation  about  the 
axis  of  maximum  rotation-area  and 
the  distance  of  the  tangent  plane 
at  the  extremity  of  the  axis  of  rota- 
tion are  invariable.  Poinsot's 
mode  of  conceiving  the  rotation 
(439i), 438-439 

764.  Permanency  of  the  instantaneous  axis 

and  of  the  axis  of  maximum  rota- 
tion-area in  the  body  (43953),  (4406), 

439-440 

765.  Surfaces  of  the  instantaneous  axes 

in  space  (442,),      .         .         .   440-442 

766.  The    velocity    of   the   instantaneous 

axis  in  the  body  (44 215),  .         .442 

767.  Case  in  which  the  axis  of  maximum 

rotation-area  describes  the  circular 
section ;  corresponding  spiral  path 
of  the  axis  of  rotation  (442i9_27^), 

442-443 

768.  Case  in  which  the  ellipsoids  of  iner- 

tia are  surfaces  of  revolution,         .  443 


769.  The  analysis  of  this  case  may  be  ex- 

tended to  that  of  a  solid  rotating 
about  a  fixed  jjoint  without  the  ac- 
tion of  external  forces,  .         .         .  443 

770-783.  The    Gyroscope    and    the 

Top, 443-451 

770.  Motion  of  a  solid  of  revolution  about 

a  fixed  point  (444,),        .         .    443-444 
7  71.  The  rotation  about  the  axis  of  revo- 
lution is  uniform,    .         .         .         .444 

772.  The  motion  of  the  gjTOscope  (4443,), 

(4457_ig),  ....    444-445 

773.  The   motion    of  the   gyroscope   ex- 

pressed by  elliptic  integrals 
(4466,1,,),  ....    445-446 

774.  When  the  velocity  of  rotation  van- 

ishes, the  gyroscope  is  a  spherical 
pendulum,      .....  446 

775.  Case   in    which   the    g^'roscope   de- 

scribes a  horizontal  circle,      .         .446 

776.  Major  Barnard's  case  of  the  gyro- 

scope in  which  the  initial  velocity 

of  the  axis  vanishes,        .         .         .447 

777.  Case  in  which  the  azimuthal  motion 

of  the  axis  is  reversed  during  the 
oscillation,      .         .         .         .         .447 

778.  Case  in  which  the  axis  of  the  gyro- 

scope becomes  the  downward  ver- 
tical  during  the  oscillation  (44 8^), 

447-448 

779.  Case  in  which  the  axis  of  the  gyro- 

scope becomes  the  upward  vertical 
during  the  oscillation  (44819),         .  448 

780.  Case  in  which  the  velocity  of  the  axis 

vanishes  for  the  upward  vertical,   .  448 

781.  Case  in  which  the   axis   constantly 

approaches  the  upward  vertical 
without  reaching  it  (449j_i6),  .         .449 

782.  The  theory  of  the  top  (444^),  (445;), 

(449.^), 449-450 

783.  Friction  in  the  case  of  the  gyroscope. 

The  sleeping  of  the  top,  .   450-451 

784-791.  The  Devil  on  Two    Sticks 

AND  the  Child's  Hoop,     .   451-456 

784.  Theory  of  the  motion  of  the  devil 

on  two  sticks  (452s,  12),    .         .    451-452 


XXXVlll 


ANALYTICAL   TABLE    OF   CONTENTS. 


785.  The   axis  of  the   de\'il  cannot   be- 

come vertical  in  the  general  case 
(452,,),.         .         .      ''.         .         .452 

786.  Case   of  the  devil  in  which  there  is 

no  rotation-area  about  the  vertical 
axis,  and  in  which  the  axis  of  the 
devil  may  become  horizontal,    452-453 

787.  Case  of  the   devil  in  which  there  is 

no  rotation-area  about  the  vertical 
axis,  and  in  which  the  axis  of  the 
devil  cannot  become  horizontal,     .  453 

788.  Case  in  which  the  axis  of  the  devil 

may  become  horizontal  with  a  gyra- 
tion about  the  vertical  axis ;  and 
the  corresponding  case  when  it 
cannot  become  horizontal,      .   453-454 

789.  Case  in  which  the  axis  of  the  devil 

may  become  vertical,     .        .         .  454 

790.  Theory  of  the  body  rolling  upon  a 

horizontal  plane  (45520, 2i),      .    454-455 


791.  Peculiar  motion  of  the  hoop  when  it 

is  nearly  falling,     .         .         .   455-456 

792-794.  Rotary  Progression,  Nuta- 
tion, AND  Variation,        .   456-457 

792.  Definition  of  nutation,    j^rogression, 

and  variation  of  axes,     .         .         .  456 

793.  Accelerative   forces   which   produce 

nutation,  progression,  or  variation,  456 

794.  Cases  of  these  various  actions,  .         .  457 

795-797.  Rolling   and   Sliding    Mo- 
tion,        457-458 

795.  General   theory   of    rolling    motion 

(457.,,), 457 

796.  General    theory    of   sliding    motion 

(4585), 457-458 

797.  Theory  of  sliding  with  friction  ;  case 

in  which  the  sliding  disappears,  and 
the  motion  becomes  that  of  rolling,  458 


CHAPTER    XIII. 

MOTION      OF      SYSTEMS. 


798.  Principles  of  power,  translation  and 

rotation  applicable  to  all  systems,  .  458 

799.  Forces  of  difi'erent   orders,  disturb- 

ing forces  and  perturbations,  .  458-459 

800.  Division   of  the  system  into   partial 

systems,  .....  459 

801-805.  Lagrange's  Method  of  Per- 
turbations,        .        .        .  459-462 

801.  Method  of  the  variation  of  the  arbi- 

trary constants  (460.>„),  .         .    459-460 

802.  Combination  of  divers  modes  of  vari- 

ation (461o,9),  .         .         .    460-461 

803.  Derivative   of   the   disturbing  force 

with  reference  to  an  arbitrary  con- 
stant (461.i,),  (462,),       .         .    461-462 

804.  Special  case  in  which  the  arbitrary 

constants  are  the  initial  values  of 
the  variables  (46 2ii,i.,),  .         .         .462 

805.  Variation  of  the  constant  of  power 

(462o..), 462 


806-808.  IjAplace's  Method  of  Per- 
turbations,        .        .        .  462-465 

806.  Direct  integration  of  the  disturbed 

functions  which  are  equivalent  to 
the  undisturbed  arbitrary  con- 
stants,        462-463 

807.  Special  case  of  fretjuent  occurrence 

in  planetary  perturbations,     .   463-464 

808.  Perturbations  of  a  projectile,    .         .  464 

809-818.  Hansen's   Method    of   Per- 
turbations,        .         .         .    465-469 

809.  The  first  principle  of  this  method,  and 

the  expression  of  the  time  as  an  in- 
variable arbitrary  constant  (465^),  465 

810.  The  principle  of  §  809,  applied  to  the 

case  of  §808  (465i^oo),  .         .         .465 

811.  The  principle  of  §  809,  applied  to  the 

case  of  §807,  and  especially  when 
the  disturbing  force  has  a  simple, 
periodic  form  (46505),  (4660,  j),   465-466 


ANALYTICAL  TABLE  OF  CONTENTS. 


XXXIX 


812.  The  second  pi-inciple  of  tliis  method 

or  the  application  of  the  perturba- 
tions to  the  element  of  time,  so  that 
one  of  the  functions  may  involve 
no  other  element  of  perturbation 
(46605), 4G6-467 

813.  Additional  perturbation  in  the  case 

of  §812  of  any  other  function 
(467io),  .         .      '   .         .         .         .467 

814.  Other  forms  of  the  perturbation  in 

the  first  ajiproximation  (46 7„),      .  467 

815.  Case  in  which  the  function  of  §  812 

does    not    involve    the    velocities, 

467-468 

816.  817.  Case  in  which  the  initial  values 

of  the  functions  of  §§  812  and  813 
are  simply  related  to  the  arbitrary 
constants,        .....  468 

818.  The    further    development    of   the 

methods  of  perturbation  Is  reserved 
for  celestial  mechanics,  .         .   468-469 

819-824.  Small  Oscillations,     .   469-472 

819.  The  theory  of  small  oscillations  Is  re- 

duced to  the  Integration  of  a  sys- 
tem of  linear  differential  equations 
(469a;), 469 

820.  The   superposition   of  small   oscilla- 

tions,           469-470 

821.  Integration  of  the  equations  of  §  820 

(4  70ij_2,), 470 


822.  Admissible  forms  of  small  oscillations 

correspond  to  stable   elements   of 
equilibrium  (470oj),         ,         ,    470-471 

823.  Independent  elements  of  oscillation 

(471^,), 471-472 

824.  Oscillation    and   vibration    pervade 

the  phenomena  of  nature,       ..        ,472 

825-830.  A  System  moving  ix  a  Re- 
sisting Medium,         .         ,  4  72-475 

825.  Equation   for   the   determination   of 

the  Jacobian  multijiller  In  such  a 
system  (472^1),        .         .         .472-473 

826.  The  factors  of  the  multiplier  corre- 

spond to  different  laws  of  resistance,  473 

827.  Cases    in    which    the    multiplier   is 

unity, 474 

828.  The  multiplier  when  the  resistance  is 

proportional  to  the  velocity,    ,  473-474 

829.  The  multiplier  when  the  resistance 

Is  proportional  to  the  square  of  the 
velocity, 4  74 

830.  Equation  of  power  for  a  system  mov- 

ing in  a  resisting  medium  (474i5_oi),  474 

831.  Motion  of  the  centre  of  gravity  in  a 

resisting  medium  (47427,31),  (4763), 

474-475 

832.  The  rotation-area  In  a  resisting  me- 

dium (475i,i_33),        .         ,         ,         .475 


833.  The  Conclusion, 


476-477 


APPENDIX. 

Note  A.    On  the  Force  of  Moving         I  Note  B.    On  the  Theory  of  Ortho- 
Bodies,         ,        ,        .        .  479-480 1  graphic  Projections,       .        .  481 


List  of  Errata, 


483-486  I  Alpjiabetical  Index,  , 


487-496 


ANALYTIC   MECHANICS. 


CHAPTER  I. 
MOTION,  FORCE,  AND  MATTER. 


§  1.  Motion  is  an  essential  element  of  all  physical  phenomena  ; 
and  its  introduction  into  the  universe  of  matter  was  necessarily  the 
preliminary  act  of  creation.  The  earth  must  have  remained  forever 
"  without  form,  and  void ; "  and  eternal  darkness  must  have  been 
upon  the  face  of  the  deep,  if  the  Spirit  of  God  had  not  first  'hnoved 
upon  the  face  of  the  waters." 

2.  Motion  appears  to  be  the  simplest  manifestation  of  power, 
and  the  idea  of  force  seems  to  be  primitively  derived  from  the 
conscious  effort  which  is  required  to  produce  motion.  Force  may, 
then,  be  regarded  as  having  a  spiritual  origin,  and  when  it  is 
imparted  to  the  physical  world,  motion  is  its  usual  form  of  mechan- 
ical exhibition. 

3.  Matter  is  purely  inert.  It  is  susceptible  of  receiving  and 
containing  any  amount  of  mechanical  force  which  may  be  commu- 
nicated to  it,  but  cannot  originate  new  force  or,  in  any  way,  trans- 
form the  force  which  it  has  received. 

1 


2  — 


CHAPTER   II. 
MEASURE   OF  MOTION  AND   FORCE. 


MEASURE    OF   MOTION. 

§  4.  Uniform  3IoUon  is  that  of  a  body  which  describes  equal 
spaces  in  equal  times. 

5.  VclocUy  is  the  measure  of  motion.  In  the  case  of  uniform 
motion  it  is  the  distance  passed  over  in  a  given  time,  which  is 
assumed  as  the  unit  of  time,  and,  in  any  case,  it  is  at  each  instant 
the  space  which  the  body  would  pass  over,  if  it  preserved  the  same 
motion  durino;  a  miit  of  time. 

6.  If  the  space  described  by  a  body  in  the  time  t  is  denoted 
by  s,  the  expression  for  the  velocity  v  is,  in  the  case  of  uniform 
motion, 

If  the  diifcrential  is  denoted  by  d  and  the  derivative  by  Z>,  the 
expression  for  the  velocity  is,  in  any  case, 

dt  ' 


11. 

MEASURE    OF   FORCE. 

7.     Experiments   have    shown  that   the   exertion  which  is  re- 
quired to  move   any  body,  is  proportional  to   the  product  of  the 


intensity  of  the  effort  into  the  space  through  which  it  is  exerted. 
This  product  is,  then,  the  proper  measure  of  the  whole  amount 
of  force  which  is  necessary  to  the  production  of  the  motion  ; 
lone-  established  custom  has,  however,  limited  the  use  of  the 
Avord  force  to  designate  the  intemity  of  the  effort,  and  the  wliole 
amount  of  exertion  may  he  denoted  by  the  term  iwwer.  Hence,  if 
the  power  P  is  produced  by  the  exertion  of  a  constant  force  F, 
acting  through  the  space  §,  the  expression  of  the  force  is 

S 

But  if  the  force  is  variable  in  its  action,  the  expression  of  its 
intensity  at  any  point  is 

ds 

8.  It  is  found  by  observation  that  the  force  of  a  moving  body 
is  proportional  to  its  velocity.  Thus,  if  m  is  the  force  of  a  body 
wdien  it  moves  with  the  unit  of  velocity,  its  force,  Avhen  it  has 
the  velocity  v,  is  mv. 

9.  Different  bodies  have  different  intensities  of  force  when 
they  move  with  the  same  velocity.  The  mass  of  a  body  is  its 
force,  when  it  moves  with  the  unit  of  velocity ;  thus,  m  in  the 
preceding  article,  denotes  the  mass  of  the  body. 

10.  The  force  communicated  to  a  freely  moving  body,  by  a 
force  which  acts  in  the  direction  of  the  motion,  is  found  to  be  the 
product  of  the  intensity  of  the  acting  force,  multiplied  by  the 
time  of  its  action.  Thus,  if  the  mass  m,  acted  upon  by  the  con- 
stant force  F,  for  the  time  t,  in  the  direction  of  its  motion,  has 
its  velocity  increased  by  v,  the  addition  to  the  force  of  the  mov- 
ing body  is 

mv  =  Ft. 


_  4  — 

In  case  the  acting  force  is  not  constant,  the  rate  at  which  the 
force  of  the  hody  increases  is 

mDt  V  =  F. 
III. 

FOKCE    OF   MOVING    BODIES. 

11.  The  ijoiuer  2vith  trhich  a  lody  moves  is  equal  to  ilie  iwoduct  of 
one  half  of  its  mass  multiplied  If/  the  square  of  its  velocitf/. 

For  if  the  hody,  of  which  the  mass  is  m,  is  acted  upon  by 
the  force  F,  until  from  the  state  of  rest  it  reaches  the  velocity 
V,  the  power  P,  which  has  been  communicated  to  it,  and  which  it 
consequently  retains,  must,  by  (Si^)  *.and  (43),  give  the  equation 

D,P^=  mDt  V. 
The  derivative  of  P  relatively  to  t,  is  by  {2^^) 

D,P  =  D,P.D,s  =  vD,  P  =  mvD,  v. 
The  integral  of  this  equation  is 

P  =  hnt^, 

to  which  no  constant  is  to  be  added,  because  the  power  vanishes 
with  the  velocity.     (JVote  A.) 

12.  Hence  the  power  of  a  moving  body  is  equal  to  one  half 
of  the  product  of  its  force  multiplied  by  its  velocity. 


*  The  form  of  reference  here  given  is  bj  means  of  numbers,  of  which,  the  leading 
number  refers  to  the  page,  and  the  secondary  number,  which  is  printed  in  smaller 
type,  refers  to  the  place  upon  the  page,  estimated  from  the  top  of  the  page,  in  lines  of 
equal  typographic  interval.  Printed  marks,  corresponding  to  these  intervals,  accom- 
pany each  cojiy  of  the  work.  Thus,  (814)  denotes  the  equation  which  is  at  the  14th 
typographic  interval  from  the  to]>  of  the  third  page. 


13.  It  is  convenient  to  refer  the  measure  of  force  to  the 
unit  of  mass  as  a  standard.  Thus,  if  F  is  the  force  exerted  upon 
each  unit  of  mass,  the  force  exerted  upon  the  body  of  which  the 
mass  is  m,  is  mF.     With  the  F,  used  in  this  sense,  (43)  becomes 

D,v=^F. 


CHAPTER  III. 
FUNDAME^s'TAL   PRINCIPLES    OF   REST   AND   MOTION. 


TENDENCY    TO    MOTION. 

§  14.  A  system  of  moving  hoclies  may  he  regarded  mechanically  as  a 
system  of  forces  or  iwwers,  ivliich  must  he  the  exact  equivalent  of  all  the 
forces  or  iwivers  which,  hy  simultaneous  or  successive  communication  to  the 
hodies,  are  united  in  its  formation. 

This  results  from  the  inertness  of  matter,  and  its  incapacity  to 
increase,  diminish,  or  vary  in  any  way,  the  power  which  it  contains. 

15.  It  also  follows  from  its  inertness,  that  matter  yields  instan- 
taneously to  every  force,  and  cannot  resist  any  tendency  to  the 
communication  or  abstraction  of  power.  With  a  system  which  is 
at  rest,  there  can  consequently  be  no  tendency  to  the  communi- 
cation of  power. 

16.  The  tendency  of  any  body  or  system  of  bodies  to  move 
in  any  given  way  is  easily  ascertained.  It  is  only  necessary  to  sup- 
pose the  system  moved  with  the  proposed  motion  to  an  infinitesnnal 


distance.  The  product  of  the  corresponding  distance,  by  which  each 
body  of  the  sj'stem  advances  in  the  direction  in  which  each  force 
acts,  multiphed  by  the  intensity  of  the  force  is,  by  §  7,  the  corre- 
sponding power  which  the  force  communicates  directly  to  the 
body,  and  through  it  to  the  system. 

The  zvhole  amount  of  iwiver  2vhicJi  is  thus  communicated  hj  all  the 
forces  to  the  si/stem,  or  rather  Us  ratio  to  the  infinitesimal  element  of  the 
proposed  motion  is  evidentln  the  measure  of  the  tendency  of  the  system  to 
this  j^rojjosed  motion. 

It  must  be  observed  that,  when  a  body  moves  in  a  direction 
02:)posite  to  that  of  the  action  of  the  force,  the  corresponding  product 
is  neQ;ative,  and  must  be  used  with  the  ne<2;ative  sim  in  forming;  the 
algebraical  sum,  which  represents  the  whole  amount  of  power  com- 
municated to  the  system. 

17.  By  a  skilful  use  of  the  principles  of  the  preceding  sec- 
tion, all  the  elementary  tendencies  to  motion  in  a  system  may  be 
determined,  and,  therefore,  all  the  elements  of  change  of  motion  in 
the  system  which  is  actually  moving,  or  all  the  conditions  of  equi- 
librium in  the  system  which  is  at  rest.     Thus,  let 

?«i,  «?2>  wzg,  &c.,  denote  the  masses  of  a  system  of  bodies; 

Fi,  F[,  F[\  &c,  the  forces  which  act  upon  each  unit  of  ?«i; 

F2,  F2,  F2,  &c.,  the  forces  which  act  upon  each  unit  of  ;;^2; 

&c,  &c. ; 

^1?  ^!/ij  f!/Tj  &c.,  the  distances   by  Avhich  w?i  advances  in  the 

direction  of  the  forces  F^,  F[,  Fi,  &c.,  in  consequence  of 

any  proposed  motion ; 
^2?  ^1/25  ^/2'j  &c. ;  (^3,  &c.,  the  corresponding  distances  for  the 

other  bodies  and  forces  of  the  system  ; 
^',  the  sum  of  all  quantities  of  the  same  kind,  obtained  by 

changing  the  accents ; 


—  7  — 

^i,  tlie  sum  of  all   quantities  of  the  same  kind,  obtained  hy 

changing  the  underwritten  numbers  ; 
2[,  the  sum  of  all  quantities  of  the  same  kind,  obtained  by 

ail  admissible  combinations  of  both  chano-es. 

The  power  communicated  to  the  system  by  the  proposed 
motion  through  nii,  1112,  &c.,  is 

^m,F,,\f,  =  m,  {F,df,  +  Fid/[  +  &c.) 
:^m^F^d/l  =  ;;^2  {F.^d'f.^  +  F^^jl  +  &c.) 
&c.  &c. ; 

and  the  whole  power  communicated  is 

^[m,F,i)f,=  :E,2'nHF,d/, 

=  :S'm^F^df^  +  2'nhF.,df.^  +  &c. 

This  is,  therefore,  the  complete  measure  of  the  tendency  in  the 
system  to  the  proposed  motion,  or  of  the  change  of  motion  which 
the  moving  system  would  experience  in  the  direction  of  the  pro- 
posed motion.  But  by  a  simple  change  in  the  values  of  cT/j,  ^f[, 
cT/2,  cV/2,  &c.,  the  tendency  to  any  other  proposed  motion  may  be 
measured ;  and,  in  the  same  way,  all  the  elements  of  the  change  of 
m.otion  may  be  definitely  ascertained. 

II. 

EQUATIONS    OF    MOTION    AND    EEST. 

18.  If,  instead  of  the  given  forces,  each  body  were  acted  upon 
by  a  force  in  the  direction  of  its  motion,  and  of  such  an  intensity  as 
to  produce  the  exact  change  of  velocity  which  it  undergoes,  this 
new  system  of  forces  would  precisely  correspond  to  that  actually 
imparted  to  the  moving  bodies,  and  would  be  the  exact  equivalent 
of  the  given  system  of  forces.     Let 


^'i?  ^'2?  ^'39  &c.  denote  the  velocities  of  the  hoclies; 

dsi,  tV6'2,  dsg,  &c.,  the  distances  by  which,  in  consequence  of 
the  proposed  arbitrary  motion  of  the  preceding  section, 
the  bodies  advance  in  the  actual  direction  of  this  motion ; 

and  then  from  (43) 

DiVi,  Z>«?'2?  -DtVs,  &c.,  are  the  intensities  of  the  new  forces 
relatively  to  the  unit  of  mass. 

The  whole  power  communicated  by  the  new  system  of  forces 
with  the  proposed  motion  becomes,  then, 

^im^Dtt\d^Si  =  miDii\dsi  -f-  m2DiV2(^S2  -\-  &c., 

and  it  must,  therefore,  be  equal  to    the    expression    (Tis)    of  the 
power  communicated  by  the  given  forces.     Hence, 

or  by  transposition 

When  the  system  is  at  rest,  this  equation  becomes 

-Zi^ii^ifT/i  =  o. 

19.  The  equation  (8ig)  in  the  case  of  motion,  or  the  equation 
(820)  in  the  case  of  rest,  although  it  appears  to  be  a  single  equation, 
involves  in  fact  as  many  equations  as  there  are  distinct  elements  of 
motion  or  rest  in  the  system  of  bodies.  For  every  such  element 
gives  a  different  set  of  values  of  cV/i,  df{,  df^,  &c.,  ds^,  ds^,  &c.,  which, 
substituted  in  (Big)  or  (820),  produce  a  corresponding  equation. 
These  equations,  therefore,  involve  all  the  necessary  conditions  of 
motion  or  rest  in  every  mechanical  problem.  All  that  remains, 
then,  is  to  determine,  by  geometrical  analysis,  the  various  elements 
of  motion  or  rest,  and  to  integrate  and  interpret  the  algebraical 


—  9  — 

equations,  into  which  (S^g)  and  (890)  are  finally  clecomposecl.  The 
Mecanique  Anali/tiquc  of  the  ever-living  Lagrange  contains  the  general 
forms  of  investigation  with  unequalled  elegance  and  perspicuity. 
But  the  special  modes  of  analysis,  which  are  peculiarly  adapted  to 
the  illustration  and  development  of  particular  problems,  have  been 
too  much  neglected,  and  the  attention  of  3'outhful  explorers  is 
earnestly  invited  to  this  unbounded  field  of  research. 


CHAPTER   lY. 
ELEMENTS   OF  MOTION. 


MOTION    OF    TRANSLATION. 

§  20.  A  single  material  point  may  be  moved  to  an  infinitesimal 
distance  in  any  direction,  which  may  be  defined  by  either  of  the 
methods  known  to  geometers,  by  the  reference,  for  instance,  to  the 
directions  of  three  mutually  perpendicular  axes.  By  the  known 
theory  of  projections,  {Note  B,)  the  distance  by  which  the  point 
advances  in  the  direction  of  its  actual  motion,  or  in  any  other  direc- 
tion, may  be  fully  determined  from  the  distances  which  it  advances 
m  these  three  directions.  The  three  distances,  moved  in  the  direc- 
tions of  the  axes,  which  are  simply  the  projections  of  the  proposed 
motion  upon  the  three  axes,  are  the  three  independeiit  elements  of 
motion  vjhich  completely  define  the   elemeiiiary  motion  of  the  sinr/le  point. 


—  10  — 

Thus  if 

dj)  denotes  the  proj^osed  elementary  motion,  if 

■^5  ^7  P,    denote  the  anf^les  which  this  motion  makes  with  the 

three  mutually  perpendicular  axes,  called  the  axes  of  x, 
7/,  and  z,  and 
dx,  df/,  dz,  the  projections  of  dp  upon  the  axes, 

the  expressions  for  these  projections  are, 

dx  =  cos  P .  dpy 
d>/  =  cos  P .  dp, 
dsz=cos^  .dp. 
If,  in  general, 

g  denotes  the  angle  which  the  directions  ofp  and  ^  make 
with  each  other,  the  distance  by  which  the  point 
advances,  in  consequence  of  the  proposed  motion,  in 
the  direction  of/  is,  by  the  theoiy  of  projections, 

d/=  cos  ^.  dp 

^  z::!-- GOS-^  .dx-r-COS-^ .d^/ -\-COS*  .dz 

X  ^  y     ^     ^  z 

:=:  J^^COS-^  .dx  : 

^  X  ' 

in  which 

2^  denotes  the  sum  of   all  the  similar  terms  obtained  by  pro- 
ceeding from  one  axis  to  each  of  the  others. 

21.  The  most  important  of  all  the  elementary  motions  of  a 
system  of  bodies  are  those  which,  being  independent  of  the  peculiar 
constitution  of  the  system,  may  be  common  to  all  systems.     Such 


—  11  — 

motions  must  be  possible,  even  if  the  bodies  which  compose  the  sys- 
tem, do  not  change  their  mutual  positions,  but  are  so  rigidly  fixed 
that  the  whole  may  be  regarded  as  one  solid  body.  It  will  be 
shown  that  there  are  but  two  distinct  classes  of  such  motions, 
namely,  those  of  tramlaiion  and  those  of  rotation. 

22.  The  motion  of  translation  is  that  by  which  all  the  points  of 
a  body,  or  system  of  l)odies,  are  transported  through  the  same  dis- 
tance in  the  same  direction.  The  projections  of  an  elementary 
translation  upon  three  rectangular  axes  are  given  by  equations 
(IO10.12),  while  (IO21),  is  the  expression  of  the  distance  by  which  the 
system,  or  any  one  of  its  bodies,  advances  in  any  direction,  such  as 
that  of/,  by  reason  of  the  proposed  translation. 

23.  Any  numl^er  of  different  elementary  translations  may  be 
supposed  to  be  given  at  the  same  time  to  a  system,  and  the  result- 
ing motion  will  l3e  such  an  elementary  translation,  that  its  projec- 
tion, estimated  in  any  direction,  will  l^e  the  sum  of  the  projections 
of  the  elementary  translations  estimated  in  the  same  direction. 

Two  coexistent  elementary  translations  may  be  combined  geo- 
m.etrically  by  setting  off  from  any  point  two  lines  of  tlie  same 
length  with  the  elementary  motions,  and  in  the  same  direction  with 
them ;  and  if  a  parallelogram  is  described  upon  these  two  lines  as 
sides,  the  diagonal,  which  is  drawn  from  the  given  point,  will  rep- 
resent in  distance  and  direction  the  resulting  elementary  transla- 
tion. 

In  the  same  way  the  geometrical  resultant  of  the  combination 
of  three  elementary  translations  may  l:»e  represented  by  the  diago- 
nal of  a  parallelepiped  described  upon  the  lines  which  represent  the 
component  translations.  But  this  parallelepiped  vanishes  when  the 
three  lines  are  in  the  same  plane. 


—  12  — 
11. 

MOTION     OF     ROTATION. 

§  24.  The  motion  of  rotation  is  that  by  which  all  the  points  of 
a  body  or  system  of  bodies  turn  about  a  fixed  line  in  the  body, 
which  line  is  called  the  axis  of  rotation.  If  one  stands  with  his  feet 
against  the  axes  of  rotation,  and  his  body  perpendicular  to  it,  and 
faces  in  the  direction  of  the  rotation,  the  podtive  direction  of  the 
axis  of  rotation  is,  in  this  treatise,  regarded  as  lying  upon  his  right 
hand,  and  its  negative  direction  upon  his  left  hand.  It  will  be  found 
convenient  to  represent  a  rotation  geometrically  by  a  distance  pro- 
portional to  the  elementary  angle  of  rotation,  set  off  upon  the  posi- 
tive direction  of  the  axis  of  rotation  from  any  point  taken  at  pleas- 
ure in  the  axis.     If 

d^  denotes  the  elementary  angle  of  rotation,  and  r  the  distance 
of  a  point  of  the  body  from  the  axis  of  rotation  ; 

rd^  is  the  elementary  distance  through  wdiich  the  point  moves 
in  consequence  of  the  rotation. 

The  form  in  which  the  subject  of  rotation  will  be  here  pre- 
sented, is  not  greatly  modified  from  that  which  it  has  finally 
assumed  in  Poinsot's  admirable  exposition  of  the  "  Theory  of  the 
Rotation  of  Bodies,''  as  it  is  printed  in  the  additions  to  the  Commis- 
sance  des  Temps  for  1854. 

25.  When  a  hody  rotates  about  an  axis,  it  is,  in  consequence  of  this 
rotation,  simidtaneoiisly  rotating  aboid  any  other  axis  ivhich  passes  through 
the  same  point,  loith  an  angle  of  rotation  which  is  represented  by  the  pro- 
jection upon  this  neiv  axis  of  the  line  ivhich  represents  the  original  angle  of 
7'otation. 

For  by  the  angle  of  rotation  ^  about  the  axis  OA  (fig.  1),  the 


—  13  — 

point  P  of  the  axis  OB,  wliicli  is  at  the  distance 

r  =  P3f 

from  the  axis  OA,  is  moved  through  the  distance  r^.  Although 
every  point  of  the  axis  OA  is  actually  at  rest,  it  has  with  respect  to 
P,  a  relative  motion,  which  is  the  negative  of  that  of  P.  A  rota- 
tion ^'  about  the  axis  OB  gives  the  point  N  of  the  axis  OA,  which 
is  in  the  plane  drawn  through  P  perpendicular  to  OB,  and  at  the 
distance 

r'^PN 

from  the  axis  of  OB,  a  motion  through  the  distance  r'(3'  taken  nega- 
tively. This  rotation  is,  then,  the  same  with  that  w^iich  the  actual 
rotation  produces  about  the  axis  OB,  if 

or  ^  =  ^=:cosJ/Pi\^ 

o         r 

=  COS  A  OB; 

that  is,  if  <3'  is  equal  to  the  projection  of  6  upon  OB. 

26.  Three  simidtaneous  elementary  rotations  about  three  axes,  tuhich 
pass  through  the  same  ^yoint,  and  are  not  in  the  same  ^ilane,  are  equivalent 
to  a  single  rotation  ahoid  the  diagonal  of  a  parallelojnjoed,  of  lohich  the  three 
lines  represeiding  the  rotations  are  the  sides,  and  the  length  of  the  diagonal 
represeiUs  the  angle  of  elementary  rotation. 

For  the  algebraic  sum  of  the  projections  of  the  sides  of  the 
parallelepiped  upon  any  line  perpendicular  to  its  diagonal  is  zero, 
and,  therefore,  there  is  no  rotation  about  any  such  line.  Hence  the 
diagonal  is  stationary,  that  is,  it  is  the  axis  of  rotation.  The  whole 
amount  of  rotation,  being  the  sum  of  the  partial  rotations  about  the 
diag-onal  which  arise  from  the  several  rotations  about  the  sides,  is 
represented  by  the  sum  of  the  projections  of  the  sides  upon  the 


—  14  — 

diagonal,  which  is,  by  the  theory  of  projections,  equal  to  the  diago- 
nal itself. 

27.  In  the  same  way,  two  simultaneous  rotations  about  the 
sides  of  a  parallelogram  may  be  combined  into  a  single  rotation 
about  the  diagonal.  In  short,  simultaneous  cicmentari/  rotations  about 
axes  ivliich  cut  each  other  may  Jje  comhined  in  the  same  ivay  as  elemoitari/ 
translations. 

28.  To  investigate  the  distance  by  which  a  given  rotation 
causes  any  point  of  a  body  or  system  to  advance  in  a  given  direc- 
tion, as  that  of  /  ;  let 

di^  be  the  elementary  angle  of  rotation  about  the  axis  of  ]j  and 
r  the  j^erpendicular  let  fall  from  the  point  upon  the  axis 
of  rotation. 

Let  a  line  be  drawn  through  the  given  point,  parallel  to  the 
projection  of  /  upon  a  plane,  which  is  perpendicular  to  the  axis  of 
rotation,  and  let 

i)  be  the  perpendicular  let  fall  upon  this  line  from  the  point  in 

which  r  meets  the  axis  of  rotation  ;  and 
r  the   angle  which  /  makes  with  the  direction  in  which  the 
point  is  moved  by  the  elementary  rotation. 

The  distance  by  which  the  point  advances  in  the  direction 
of /is 

cT/=/cos  ^.8^  =  /cos  ^,BmP..d& 

in  which  ^   should  be  taken  positively  when  the  point  is  moved 
towards  the  positive  direction  of  /. 

29.  If  three  rectangular  axes  are  drawn  through  any  point 
of  the  axis  of  rotation,  and  if 


—  15  — 

^t?^,  d^y,  d^^  are  the  projections  of  d&  upon  these  axes,  the  dii^- 
tance  hy  which  the  point  {x,  ?/,  z)  is  moved  in  the  direc- 
tion of  the  axis  of  x,  is 

dx  =  ijd^^  —  zd^y 

=  (ycos^  —  0cos^)  (5(5 

=  (cos  ^  cos  ^  —  COS  ^  cos  ^)  r()'i5 

=  (cos  '^  cos-?^  —  COS  ^  COS  ^^)  coscc  **  .r'(5"^ 


=:COS^/(^^ 


=  (C0S^C0S^  COS      COS  ^  )?■'()' t5. 

\      y       z  z        y' 


There  are  similar  expressions  for  the  distances  by  which  the 
point  advances  in  the  directions  of  the  axes  of  y  and  z,  which  may 
be  found  by  advancing  each  of  the  letters  .r,  y,  z^  and  x  to  the  fol- 
lowino;  letter  of  the  series. 

30.  The  two  last  members  of  equation  {Vo-^  divided  by  r'(5't5 
give  the  following  theorem ; 

cos  ^  =  cos  ^  cos  ^^  —  cos  ^  cos^, 
X  y        ^  ^        y 

in  which  the  direction  of  ^  is  that  of  the  perpendicular  to  the  com- 
mon plane  of  r  and  p,  and  it  is  taken  upon  that  side  of  the  plane 
for  which,  a  positive  rotation  about  it,  would  correspond  to  a 
motion  through  the  acute  angle  from  r  to  p. 

31.  If  there  were  another  system  of  rectangular  axes,  o',y\ 
and  /,  equation  (ISso)  applied  to  them  would  give 

cos       :=  COS  ^  COS       COS  -^  COS      . 

X  y  ^  z  y 

In  this  equation  each  of  the  letters  x,  y,  z,  and  x  might  be 
advanced  to  the  subsequent  letter  of  the  series,  as  well  as  each  letter 


IG 


of  the  series  x' ,  y',  z,  and  x'.     In  this  way  eight  other  equations 
might  be  found  simihir  to  equation  (ISss). 


III. 

COMBINED    MOTIONS    OF    ROTATION    AND    TRANSLATION. 

32.  An  elementary  rotation,  comhined  ivith  an  elementary  translation 
in  any  direction,  tvJiich  is  iderpendicular  to  the  axis  of  rotation,  is  equivalent 
to  an  equal  elementary  rotation  about  an  axis  ivhich  is  parallel  to  the  origi- 
nal axis  of  rotation.  The  position  of  the  new  axis  is  determined  hy  the  con- 
dition that  each  of  its  points  is  carried  hy  the  original  elementary  rotation  as 

far  as  by  the  elementary  translation,  but  in  an  opposite  direction. 

For  the  given  motions  cancel  each  other's  action  upon  each 
point  of  the  new  axis,  and  leave  it  stationary ;  while  the  original 
axis  advances  with  the  elementary  translation  by  the  exact  dis- 
tance which  corresponds  to  the  elementary  rotation  about  the  new 
axis.  The  common  plane  of  the  two  axes  is  perpendicular  to  the 
direction  of  the  translation. 

33.  Any  simultaneous  elementary  rotations  about  axes  parallel  to  each 
other  are  equivalent  to  a  single  rotation,  equal  to  their  sum,  and  about  an  axis 
parallel  to  the  given  axes,  combined  ivith  an  elementary  translation  equal  to 
the  motion  ivhich  any  point  of  the  nciu  axis  receives  from  their  simultaneous 
action. 

This  is  a  simple  deduction  from  the  preceding  proposition. 

34.  Let  there  be  three  rectangular  axes,  such  that  the  new 
axis  of  rotation  may  be  that  of  s  ;  let 

.^1,^1,  rr2,^2j  &c.,  be  the  points  in  which  the  original  axes  cut 

the  plane  of  xy  ;  and  let 
(T^i,  (5"^25  &c.,  be  the  elementary  angles  of  rotation  about  these 

axes. 


—  17  — 
The  elementary  rotation  about  the  axis  of  z  is 

The  elementary  translations  in  the  directions  of  the  axes  of  x 
and  y  are  by  {Vl^^ 

d^o  =  — -i-'^'i^^^i- 
The  distances  through  which  anj^  point  (.r,  ?/,  z)  is  carried  for- 
ward in  the  directions  of  the  axes,  are 

d//  =^  d'^Q  -\-xd^  z=:^  —  ^1  Xi  di^-^-\-  X  -2*1  (J  ^1. 
The  points  are,  therefore,  at  rest  for  which 

0  =  dxQ  —  ydi^  =  ^i,yicT^i — y  JE-^^d&i, 
0  =  (5>o  +  ^■f'>^^  =  —  ^1  -^1  (^^1  +  x2:j&^. 

These  are,  therefore,  the  equations  of  the  axis  of  rotation,  an  elementary 
rotation  about  tvhich,  equal  to  the  sum  of  all  the  elementary  rotations,  is 
equivalent  to  the  conihination  of  all  the  elementary  rotations. 

35.  If  the  original  elementary  rotations  are  all  equal,  and  if 
there  are  n  axes  of  rotation,  the  equations  (172)  and  (17ii)  become 

(5(5  -z^ind^y, 

(3\r=(^i^i  — 7?//)  (5^^i, 
dy  =  ( —  ^1  x^  -\-  n .?;)  d  ^i . 

The  equations  (17i6)  give  for  the  single  axis  of  rotation 


V 

y=- 


n    ■ 


36.     If  any  of  these  rotations  are  about  an  axis  lying  in  the 
opposite  to  the  assumed  direction,  they  may  be  regarded  as  nega- 

3 


—  18  — 

tive  rotations  about  axes  having  the  same  direction  as  the  assumed 
one,  and  may  be  combined  algebraically  in  the  preceding  sums. 

37.  When  the  second  member  of  equation  (172)  vanishes,  the 
resulting  rotation  disappears,  and  the  given  elementary  rotations 
are  equivalent  to  the  elementary  translation  defined  by  equa- 
tions (176). 

38.  Two  equal  rotations  about  axes,  which  are  parallel,  but 
have  opposite  directions,  constitute  a  combination  which  Poinsot 
has  called  a  couple  of  rotations. 

A  couple  of  elementmyj  rotations  is,  therefore,  equal  to  an  elementary 
translation  in  a  direction  perpendicular  to  the  common  plane  of  the  axes, 
and  equal  to  the  product  of  the  distance  between  the  axes  multiplied  b?j  the 
elemoitary  angle  of  rotation. 

39.  Any  simultaneous  elementary  motions  of  rotation  and  translation 
are  equivalent  to  a  single  elementary  rotation  about  an  axis,  combined  ivith 
an  elementary  translation  in  the  direction  of  the  axis  of  rotation. 

For  each  rotation  may  be  resolved  into  a  translation  and  a 
rotation  about  an  axis  passing  through  any  assumed  point.  But  all 
the  elementary  rotations  about  axes  passing  through  the  same  point 
are  equivalent  to  a  single  rotation  about  an  axis  passing  through 
the  point,  and  all  the  translations  are  equivalent  to  a  single  transla- 
tion. The  single  translation  may  be  resolved  into  two  translations, 
of  which  one  is  parallel,  and  the  other  perpendicular  to  the  single 
axis  of  rotation.  The  translation,  which  is  perpendicular  to  the 
axis  of  rotation,  combined  with  the  rotation,  is  equivalent  to  a  sin- 
gle rotation  about  an  axis,  parallel  to  the  single  axis,  and,  therefore, 
having  the  same  direction  with  the  remaining  translation. 

40.  Every  possible  motion  of  a  rigid  system  or  body  is  equivalent  to 
a  combination  of  the  motions  of  translation  and  rotation. 

This  is  evident,  if  it  can  be  shown  that,  by  such  a  combination 
of  motions,  any  three  points.  A,  B,  and  C,  of  the  system,  can  be  car- 


—  19  — 

ried  to  any  positions,  A',  B',  and  C,  in  which  it  is  possible  for  them 
to  be  placed.  For  tliree  points  of  a  rigid  system  not  in  the  same 
straight  line  completely  determine,  by  their  position,  that  of  the 
whole  system.  Now,  by  a  translation  of  the  system,  equal  to  that 
by  which  A  might  be  directly  moved  from  A  to  A,  the  point  A  is 
actually  brought  to  the  position  A.  By  a  subsequent  motion  of 
rotation  about  an  axis,  which  is  perpendicular  to  each  of  the  lines 
AB  and  A  B',  the  point  i?  may  be  moved  to  B' ;  and  then  by  a 
rotation  about  AB'  the  point  0  may  be  carried  to  C.  Hence  the 
whole  motion  is  accomplished  by  one  translation  and  two  rotations. 
Every  elementary  motion  of  a  rigid  system  must  then  be 
equivalent  to  a  single  rotation  about  an  axis  and  a  translation  in 
the  direction  of  the  axis  of  rotation.  This  motion  is  perfectly  rep- 
resented by  that  of  the  screw,  whose  helix  causes  it  to  advance  in 
the  direction  of  the  axis  about  which  it  is  turiiiuy:;. 

41.  During  each  instant  of  its  motion,  a  rigid  system  rotates 
about  an  axis,  which  is  called  the  mstanfaneous  axis  of  rotation.  This 
axis  is  generally  varying  its  positio.i  in  the  system  and  in  space 
from  one  instant  to  another,  which  renders  it  difficult  to  form 
a  distinct  conception  of  the  nature  of  the  corresponding  motion  of 
the  system. 

42.  In  attempting  to  conceive  of  the  motion  of  a  rigid  system, 
it  is  expedient,  at  first,  to  neglect  the  translation  in  the  direction  of 
the  axis  of  rotation,  and  to  assume  that  the  motion  is  solely  that 
of  rotation.  The  successive  positions  of  the  axis  of  rotation  in  the 
system  form  by  their  union  a  surface  which  turns  with  the  system ; 
and  its  successive  positions  in  space  form  another  fixed  surface.  In 
the  motion  now  considered,  the  moving  surface  rolls  on  the  fixed 
surface  without  sliding,  and  carries  the  system  with  it. 

43.  If  the  axis  of  rotation  does  not  move  perpendicularly  to 
itself  each  of  these  surfaces  is  evidently  a  developable  surface,  and 


—  20  — 

in  the  act  of  rolling  the  line  of  retrogression  of  the  one  falls  npon 
that  of  the  other ;  so  that  these  two  lines  are  of  the  same  length. 
Upon  the  surfaces,  developed  into  a  plane,  the  two  lines  of  retro- 
gression will  be  precisely  alike. 

In  combinins:  with  this  rotation  the  translation  in  the  direction 
of  the  axis  of  rotation,  the  surface,  generated  by  the  instantaneous 
axis  in  the  moving  system,  remains  unchanged.  But  the  fixed  sur- 
face, generated  by  the  instantaneous  axis,  is  changed ;  it  is  still  a 
developable  surface  obtained  from  that  in  which  the  translation  is 
neglected,  by  adding  to  each  element  of  the  arc  of  the  curve  of 
retrogression,  the  elementary  translation  in  the  direction  of  the  axis 
of  rotation.  In  the  actual  motion,  the  moving  surflice  rolls  upon 
the  fixed  surface,  and  glides  simultaneously  in  the  direction  of  the 
line  of  contact,  so  as  to  keep  the  curves  of  retrogression  constantly 
in  contact. 

In  this  general  case,  the  whole  length  of  the  arc  of  the  fixed 
curve  of  retrogression  is  equal  to  that  of  the  moving  curve  aug- 
mented by  the  whole  amount  of  translation  in  the  direction  of  the 
axis  of  rotation. 

When  the  elementary  translation  is  equal  to  the  elementary 
arc  of  the  moving  curve  of  retrogression,  but  lies  in  the  opposite 
direction,  there  is  a  corresponding  cusp  in  the  fixed  curve  of  retro- 
gression. 

A  point  of  inflection  in  the  curves  of  retrogression  generally  cor- 
responds to  a  change  in  the  direction  of  the  rotation.  A  similar 
combination  of  the  translation  with  the  rotation  can  be  introduced 
into  the  general  case  of  motion. 

44.  When  either  of  the  surfaces  of  the  instantaneous  axis  is 
a  cone,  the  curve  of  retrogression  is  reduced  to  a  point  which  is  the 
vertex  of  the  cone.  When  both  of  the  surfaces  are  cones,  there  is  no 
translation  in  the  direction  of  the  axis. 


—  21  — 

When  either  of  the  surfaces  is  a  cjdinder,  both  surfaces  must 
be  cylinders;  and  the  lines  of  retrogression,  removing  to  an  infinite 
distance,  cannot  be  used  for  guiding  the  motion  of  translation. 
But  in  this  case,  a  section  may  be  made  of  one  of  the  cylinders  per- 
pendicular to  its  axis,  and  in  the  actual  motion  the  moving  cylinder 
will  move  so  as  to  keep  the  point,  in  which  the  perimeter  of  this 
section  touches  the  other  cylinder,  upon  a  curve  proj)erly  drawn 
upon  that  cylinder. 

45.  The  general  motion  of  a  rigid  system  may  be  conceived  as 
a  translation,  equal  to  that  of  any  one  of  its  points  assumed  at  will, 
combined  with  a  rotation  about  an  instantaneous  axis  of  rotation 
passing  through  the  point.  If  the  translation  is  neglected,  the  rota- 
tion is  effected  as  in  §  42  by  rolling  a  cone,  of  which  the  assumed 
point  is  the  vertex,  and  which  carries  the  system  with  it,  in  its 
motion,  about  a  fixed  cone,  of  which  the  same  point  is  the  vertex. 
The  translation  may  be  simultaneously  effected  by  moving  the  two 
cones  in  space,  with  a  translation  equal  to  that  which  belongs  to 
their  vertex  in  the  actual  motion  of  the  system. 

46.  For  all  the  points  of  the  instantaneous  axis  in  each  of  its 
positions,  the  corresponding  centres  of  greatest  curvature  of  either 
of  the  conical  surfaces  which  it  describes,  are  all  upon  the  same 
straight  line  passing  through  the  vertex. 

In  the  case  of  the  right  cone,  or  of  the  right  cylinder,  the  axis 
of  revolution  is  the  line  of  the  centres  of  greatest  curvature.  In  all 
these  investigations  the  plane  may  be  regarded  either  as  a  cylinder 
of  infinite  radius,  or  as  a  cone,  of  which  the  angle  at  the  vertex  is 
equal  to  two  right  angles. 

47.  The  elementary  rotation  of  the  system  may  be  conceived 
as  deconiposed  into  two  elementary  rotations  about  the  lines  of  the 
centres  of  greatest  curvature  as  axes  of  rotation.  By  the  rotation 
about  the  line,  which  unites  the  centres  of  the  fixed  surface,  the 


—  22  — 

instantaneous  axis  receives  its  elementary  motion  in  space,  and  is 
carried  to  its  proper  position  upon  tlie  fixed  surface.  By  the  rota- 
tion about  the  Une  wliich  unites  the  centres  of  the  moving  surface, 
the  system  receives  that  additional  rotation  which  is  required  to 
turn  the  moving  surface  into  that  position  in  which  it  may  have  the 
proper  line  of  contact  with  the  fixed  surface.  Each  of  these  rota- 
tions produces  a  sliding  of  the  moving  upon  the  fixed  surface  ;  but 
as  the  sliding  produced  by  the  one  is  just  equal  and  opposite  to  that 
]3roduced  by  the  other  rotation,  the  two  rotations  cancel  each 
other's  action  in  this  respect,  and  there  is  no  sliding  in  the 
combined  motion,  but  a  simple  rolling  of  one  surface  uj)on  the 
other. 

48.     Let 

«y  be  the  acute  angle  which  the  instantaneous  axis  of  rota- 
tion makes  with  the  line  of  the  centres  of  curvature 
of  the  fixed  surface  ; 

«„,  that  which  it  makes  with  the  line  of  the  centres  of  cur- 
vature of  the  moving- surface,  this  angle  heing  positive 
when  the  two  lines  of  the  centres  are  on  opposite 
sides  of  the  instantaneous  axis,  and  negative,  when 
they  are  upon  the  same  side  ; 

dm  the  elementary  angle  by  which  the  instantaneous  axis 
changes  its  direction ; 

(5"  ^f  the  elementary  angle  of  rotation  about  the  line  of  cen- 
tres of  the  fixed  surface  ;  and 

d  (5,„  the  elementary  angle  of  rotation  about  the  line  of  cen- 
tres of  the  moving  surface. 

Since  the  instantaneous  axis  must  be   carried  forward  by  the 
rotation  about  the  fixed  axis,  and  backward  by  the  rotation  about 


23 


the  moving  axis  just  as  far  as  its  actual  change  of  position,  its  ele- 
mentary angle  of  change  of  direction  is 

d(i)  =id  &j:  sin  «/=  d  ^,„ .  sin  a„^ . 

But  the  combination  of  the  two  rotations  about  these  axes 
gives  the  actual  rotation  about  the  instantaneous  axis,  and  there- 
fore, 

d"  ^  =  (5"  <5y.  cos  c(f-\-^  ^m  •  COS  a^ 
=^  (cot  «/-|-  cot  «„,)  d  ci» 

sin  («/+«,„)  V 
Sin  Ujr  sin  «,„ 

49.  When  the  surfaces  described  by  the  instantaneous  axis  are 
cylinders,  let 

()j-  and  ()„j  be  the  respective  radii  of  greatest  curvature  of  the 
fixed  and  moving  surfaces  at  any  point  of  their  mutual 
contact ;  and 

djy  the  elementary  distance  which  the  instantaneous  axis  moves 
in  a  direction  perpendicular  to  itself 

The  conditions  of  the  motion  of  the  instantaneous  axis  give  the 
equations 

dp  =  Qyd  <5^=  ±  Q,n  ^  ^\n  ; 

in  wdiich  the  upper  sign  corresponds  to  the  case  where  the  lines  of 
the  centres  of  curvature  are  upon  opposite  sides  of  the  instanta- 
neous axis,  and  the  lower  sign  to  that  in  which  they  are  upon  the 
same  side.     The  rotation  about  the  instantaneous  axis  is 


d6  =  dd^+di\. 


—  24  — 
lY. 

SPECIAL   ELEMENTS    OF    MOTION   AND    EQUATIONS    OF    CONDITION. 

50.  The  variation  of  each  independent  element  of  position  of 
a  system  gives  an  independent  element  of  motion.  But  the  ele- 
ments of  position  are  various,  and  must  be  selected  in  each  case 
with  special  reference  to  the  problem  under  discussion.  It  often 
occurs  that  parts  of  the  system  are  rigidly  connected ;  such  parts 
are  themselves  rigid  systems,  and  subject  only  to  motions  of  trans- 
lation and  rotation,  and,  therefore,  none  but  such  elements  are 
required  for  the  investigation  of  their  motions. 

Points  of  the  system  are  sometimes  restrained  to  move  upon 
given  surfaces,  and,  in  this  case,  it  may  be  expedient  to  introduce 
elements  of  position  dependent- upon  the  principal  lines  of  curva- 
ture of  these  surfaces,  or  elements,  in  reference  to  which  the  sur- 
faces are  peculiarly  simple  or  symmetrical.  Points  of  the  system 
may  be  compelled  to  preserve  simple  geometrical  relations  to  each 
other,  which  may  suggest  appropriate  elements  of  position  to  the 
skilful  analyst ;  or  he  may  find  indications  to  direct  his  choice  in 
the  very  nature  of  the  motion  itself 

51.  It  is  often  desirable  to  adopt  a  combination  of  elements 
of  position  which  are  not  wholly  independent  of  each  other,  but  are 
subject  to  certain  mutual  restrictions.  These  restrictions,  when 
they  are  expressed  algebraically,  are  called  equations  of  condition. 
They  may  assume  the  differential  form  of  ecjuations  between  the 
elementary  motions ;  or  they  may  be  finite  equations  between  the 
elements  of  position,  in  which  case  they  may  be  reduced  by  differ- 
entiation to  equations  between  the  elementary  motions. 

By  means  of  the  equations  of  condition,  as  many  of  the  ele- 
ments of  motion  may  be  determined  in  terms  of  the  rest  as  there 


—  25  — 

are  equations  of  condition ;  and  the  remaining  elementary  motions 
may  be  regarded  as  independent  of  each  other. 

52.  Instead  of  introducing  into  the  equations  (Sig)  and  (820)  of 
motion  and  rest  the  special  values  of  ds^,  ds2,  &c.,  cT/*i,  ^f^,  kc,  for 
each  particular  element  of  motion,  their  general  values  may  be 
found  in  terms  of  all  these  elements.  When  the  elementary 
motions  are  wholly  independent,  their  coefficients  in  these  equa- 
tions give,  when  they  are  equalled  to  zero,  the  same  equations 
which  would  have  been  obtained  by  the  special  investigations. 
But  when  the  elements  are  not  independent,  all,  except  the  inde- 
pendent elements  can  be  eliminated  by  means  of  the  values  given 
by  the  equations  of  condition. 

The  equations  (Sig)  and  (820)  of  motion  and  rest,  on  account  of 
their  differential  form,  are  necessarily  linear  in  reference  to  the  ele- 
mentary motions  ;  and  the  differential  equations  of  condition  are 
likewise  linear.  The  proposed  elimination  may  therefore  be  con- 
ducted by  the  mdJiod  of  multipliers.  By  this  process  each  differential 
equation,  multiplied  by  an  unknown  quantity,  is  to  be  added  to  the 
given  equation  of  motion  or  rest.  The  unknown  multipliers  are  to 
be  determined  by  the  conditions  that  the  coefficients  of  the  elemen- 
tary motions,  which  are  to  be  eliminated,  become  equal  to  zero. 
Since  the  remaining  elementary  motions  are  indejDcndent  of  each 
other,  their  coefficients  must  also  be  equalled  to  zero.  In  the  sum, 
therefore,  obtained  by  the  addition  of  the  equations,  each  of  the 
coefficients  of  the  elementary  motions  is  equal  to  zero.  The  num- 
ber of  unknown  quantities  is  increased  in  this  process  by  that  of  the 
unknown  multipliers ;  but,  because  there  are  as  many  equations  of 
condition  as  there  are  multipliers,  the  whole  number  of  equations, 
including  the  equations  of  condition,  in  their  finite  form,  is  just 
sufficient  to  determine  the  values  of  the  multipliers  and  of  all  the 
elements  of  position. 

4 


—  26  — 

53.  Let 

l3e  one  of  the  equations  of  condition  in  its  finite  form ;  and  let  its 
differential  form  138 

dL^  =  0. 

Let  also, 

I  be  tlie  unknown  multiplier  by  which  it  is  to  be  multiplied. 

The  sum  obtained  by  adding  the  similar  products  of  all  the  equa- 
tions of  condition  to  equation  (S^g)  or  (820)  is 

which  is  the  equation  of  motion  or  rest,  and  in  which  the  general 
values  of  ds^,  (T/j,  &c.,  are  to  be  substituted,  and  the  coefficient  of 
each  elementary  motion  is  to  be  equalled  to  zero. 

54.  Each  equation  of  condition  becomes  the  equation  of  a 
surface,  to  which  any  one  of  the  points  w^hose  elements  of  position 
occur  in  the  equation  is  restricted,  provided  that,  for  the  moment, 
the  variations  of  all  the  other  elements  are  neglected.  Since  the 
point  is  restricted  to  move  upon  the  surfiice,  it  cannot  move  in  the 
direction  of  the  normal  to  the  surface.  Let  a  system  of  three  rec- 
tangular axes  be  adopted,  and  let 

i\^be  the  normal  to  the  surface. 

Its  variation,  arising  from  the  variation  of  coordinates,  which  may 
be  regarded  as  the  elements  of  position  of  the  point,  is 

If  the  equation  of  the  surface  is  (263),  Avith  the  omission  of  the  num- 


—  27  — 

bers  written  below,  which  may  be  neglected  in  the  general  discus- 
sion, its  variation  is 

Let,  then, 

and  the  angle,  made  by  the  normal  with  one  of  the  axes,  is  given 

by  the  equation 

X  _  D,L 

COSy -JJ-) 

which  substituted  in  (2629)  gives 


dX= 


M         ~  M  ' 


Hence  the  equation  of  condition  with  its  multiplier  ma}-  be  writ- 
ten in  the  form 

WL  =  ).3IdX=0; 

and  this  form  may  be  substituted  in  the  equations  (2612)  and  (2G13) 
of  motion  and  rest. 


—  28 


CHAPTER   V. 
FORCES   OF   NATURE. 


I 

THE   POTENTIAL   AND    ITS    RELATIONS   TO    LEVEL    SURFACES,    THE    POSITIONS    OF 
EQUILIBRIUM,   AND    THE   POSSIBILITY    OF   PERPETUAL    MOTION. 

§  55.  It  appears,  at  first  sight,  to  be  inconsistent  with  the 
assumed  spiritual  origin  of  force,  that  the  principal  forces  of  nature 
reside  in  centres  of  action,  which  are  not  thinking  beings,  but  parti- 
cles of  matter.  The  capacity  of  matter  to  receive  force  from  mind 
in  the  form  of  motion,  contain  and  exhibit  it  as  motion,  and  commu- 
nicate it  to  other  matter,  under  fixed  laws,  is  not,  however,  less  diffi- 
cult or  more  conceivable  than  the  capacity  to  receive  and  contain  it 
in  a  more  refined  and  latent  form,  from  which  it  may  become  mani- 
fest under  equally  fixed  laws.  It  is  only,  indeed,  when  force  is  thus 
separated  from  mind,  and  placed  beyond  the  control  of  will,  that  it 
can  be  subject  to  precise  laws,  and  admit  of  certain  and  reliable 
computation. 

56.  The  laws  of  the  development  of  power  in  nature  are  of 
two  classes.  In  the  one  class,  the  forces  depend  solely  upon  the 
relative  positions  of  the  bodies,  and  may  be  called  fixed.  In  the 
other  class,  the  forces  depend,  not  only  upon  the  positions  of  the 
bodies,  but  also  upon  their  actual  state  of  power,  especially  upon 
the  velocities  and  directions  of  their  motions  \  and  these  forces  may 
be  called  vtiricMc. 

57.  The  most  fruitful  and  enlarged  view  of  the  fixed  forces  of 


—  29  — 

nature,  and  one  which  peculiarly  corresponds  to  their  laws  of  action 
so  fiir  as  they  have  been  observed,  is  to  regard  them  as  the  mani- 
festations of  the  di/namic  situation  of  the  bodies  which  exhibit  them. 
The  dynamic  situation  depends  solely  upon  the  masses  and  posi- 
tions of  the  bodies;  it  is  a  condition  oi^  form,  and  its  research  is  a 
problem  of  pure  geometry.  The  algebraic  function  which  embodies 
the  idea  of  the  dynamic  state  is  called  the  potential.  Its  complete 
investigation  and  determination  involves  the  solution  of  all  the 
problems  which  can  arise  in  regard  to  the  power  and  the  conditions 
of  force  of  all  systems,  whether  they  are  at  rest  or  in  motion,  so  far 
at  least  as  the  fixed  forces  of  nature  are  concerned. 

The  amount  of  power  of  a  system  is  not  to  be  inferred  from  its 
situation,  although  there  is  a  certain  measure  of  power  appropriate 
to  that  situation.  It  is  this  latter  power  which  is  expressed  by  the 
potential  of  the  system,  and  expressed  as  a  function  of  all  the  ele- 
ments of  position,  by  which  the  situation  is  defined. 

58.  TJic  power  of  a  moving  system  increases  or  decreases  vitJi  the 
poiver  tvMch  t>elongs  to  its  situation,  and  tlie  increase  or  decrease  of  its  power 
is  measured  hj  that  of  its  potenticd. 

59.  Hence,  if  a  sj^stem  moves  from  a  state  of  rest,  its  power  is 
constantly  equal  to  the  excess  of  its  potential  over  the  initial  value 
of  the  potential ;  and  it  can  never  arrive  at  a  position  in  which  the 
potential  would  be  less  than  its  initial  value.  No  system,  indeed, 
can  move  to  a  situation  in  which  the  potential  would  be  diminished 
more  than  the  initial  power  of  the  system. 

60.  When  a  system  is  in  a  permanent  state  of  rest  which  the 
actual  forces  do  not  tend  to  disturb,  its  dynamic  condition  is  such, 
that  the  power  of  the  system  is  not  changed  by  a  slight  change  of 
position.     Hence, 

The  potenticd  of  a  system  tvhich  is  in  equilibrium,  is  genercdly  a  maxi- 
mum or  a  minimum.     The  exceptional  case  of  a  condition  of  indiffer- 


—  30  — 

ence  rarely  occurs  in  nature ;  but  even  this  case  may  be  philosojDhi- 
cally  regarded  as  the  combination  of  a  maximum  and  minimum,  or 
as  the  result  of  several  such  combinations. 

61.  When  a  moving  system  passes  through  a  position  of  equi- 
librium, or  a  position  which  is  one  of  equilibrium  in  reference  to 
the  element  of  position  Avith  which  the  system  is  changing  its  place, 
the  jDower  of  the  system  is  either  a  maximum  or  a  minimum,  or  in 
a  condition  of  indifference. 

62.  When  a  system,  in  a  state  of  rest,  is  placed  very  near  the 
position  of  equilibrium,  it  cannot  tend  to  move  away  from  the  posi- 
tion of  equilibrium,  if  the  potential  of  that  situation  is  a  maximmn 
relatively  to  the  element  by  which  the  system  is  removed  from 
it ;  and  it  cannot  tend  to  move  towards  the  situation  of  equili- 
brium, if  the  potential  is  a  minimum  for  the  same  element.  On 
this  account  the  equilibrium  is  stable,  in  reference  to  those  elements 
for  which  the  potential  is  a  maximum,  and  it  is  iinstahle  in  reference 
to  these  elements,  for  which  the  j)otential  is  a  minimum. 

63.  As  when  a  function  changes  in  consequence  of  the  change 
of  any  one  of  its  variables,  the  maxima  and  minima  succeed  each 
other  alternately ;  in  the  motion  of  a  system,  the  positions  of  stable 
and  unstable  equilibrium,  relatively  to  the  element  of  change  of 
position,  succeed  each  other  alternately.  Situations  of  equilibrium 
of  indifference  may  be  interposed  without  disturbing  the  order  of 
succession  of  the  situations  of  stable  and  unstable  equilibrium.  If 
the  system  returns  to  its  initial  position,  it  must  have  passed 
through  an  even  number  of  such  situations  of  equilibrium,  rela- 
tively to  the  element  of  change  of  position,  half  of  which  must  have 
been  positions  of  stable,  and  the  other  half  positions  of  unstable 
equilibrium.  In  general,  these  situations  will  not  be  positions  of 
absolute  equilibrium,  but  only  such  in  reference  to  the  changing 
element  of  motion. 


—  31  — 

64.  Fixed  forces  might  easily  be  imagined  different  from 
those  of  nature,  and  in  the  action  of  which  the  power  of  a  moving 
system  would  depend  upon  its  previous  situations  as  well  as  upon 
its  actual  position.  "With  such  forces  the  increase  or  decrease  of 
power  of  a  s^'stem  would  vary  with  the  path  which  it  pursued  in 
moviuo;  from  one  situation  to  another,  and  would  be  greater  bv  one 
path  than  by  another.  The  change  of  power  for  each  element  of 
any  given  path,  would  still  be  computed  by  the  process  of  §  IT, 
and  thence  the  whole  change  of  power  would  be  obtained  by  inte- 
gration. If  the  motion  of  the  system  were  reversed,  and  it  were 
carried  back  through  the  same  path  to  its  initial  position,  its  initial 
power  would  be  restored.  If,  of  two  courses,  by  which  a  system 
could  move  from  one  situation  to  another,  it  were  forced  to  go  by 
that  through  which  it  would  arrive,  with  the  greater  power  at  its 
final  position,  and  if  it  were  then  made  to  return  to  its  initial  posi- 
tion by  the  other  path,  it  would  return  with  an  increased  power ; 
if  it  were  afj;ain  to  move  throuu'h  the  same  circuit,  it  would  a<2:ain 
return  with  an  equal  additional  increase  of  power ;  and,  by  succes- 
sive repetitions  of  this  process,  the  power  might  be  increased  to  any, 
even  to  an  infinite  amount.  Such  a  series  of  motions  would  receive 
the  technical  name  of  a  perpetual  motion,  by  which  is  to  be  under- 
stood, that  of  a  system  which  would  constantly  return  to  the  same 
position,  with  an  increase  of  power,  unless  a  portion  of  the  power 
were  drawn  ofi"  in  some  way,  and  appropriated,  if  it  were  desired,  to 
some  species  of  work.  A  constitution  of  the  fixed  forces,  such  as 
that  here  supposed,  and  in  which  a  perpetual  motion  would  be  pos- 
sible, may  not,  perhaps,  be  incompatible  with  the  unbounded  power 
of  the  Creator ;  but,  if  it  had  been  introduced  into  nature,  it  would 
have  proved  destructive  to  human  belief,  in  the  spiritual  origin  of 
force,  and  the  necessity  of  a  First  Cause  superior  to  matter,  and 
would  have  subjected  the  grand  plans  of  Divine  benevolence  to 
the  will  and  caprice  of  man. 


net 

oJ   — 

65.  A  surface,  for  each  of  whose  pomts  the  potential  has  the 
same  value,  may  be  called  a  level  surface.  A  level  surface  may  be 
drawn  through  any  point  in  space. 

Since  the  potential  of  every  finite  system  of  nature  vanishes 
for  an  infinitely  distant  point,  all  the  level  surfaces  of  nature  are  finite, 
and,  returning  into  themselves,  include  a  space  tvhich  they  tvholli/  surround, 
ivitli  the  exception  of  those  level  surfaces  for  tvhich  the  potential  is  zero. 

66.  A  material  point,  placed  upon  a  level  sm^face,  has  no  ten- 
dency to  move  in  the  direction  of  the  surface,  because  there  is  no 
increase  of  power  in  such  direction.  The  tendencij  of  a  material  point 
to  motion  is,  therefore,  'perpendicular  to  the  level  surface  upon  ivhich  it  is 
placed. 

67.  If  two  level  surfaces  are  drawn  infinitely  near  to  each 
other,  a  material  point,  flaced  upon  cither  of  them,  tends  to  move  in  the 
direction,  from  the  surface  of  the  less  potential  towards  the  other,  tuith  a 

force  ivhich  is  measured  hj  the  quotient  of  the  difference  of  the  potentials  of 
the  two  surfaces,  divided  hj  their  distance  apart. 

Hence,  if  the  surfaces  are,  throughout,  at  the  same  distance 
apart,  the  disposition  to  motion  is  everywhere  the  same. 

If  the  surfaces  w^ere  to  intersect  each  other,  the  tendency  to 
motion  in  the  line  of  intersection  would  be  infinite  ;  but,  since  there 
is  no  such  infinite  tendency  to  motion  in  nature,  each  level  surface  of 
nature  must  he  whollij  included  within  every  other  level  surface,  ivithin  ivhich 
any  portion  of  it  is  included.  For  the  same  reason,  the  potential  in  nature 
is  always  a  continuous  function. 

68.  Within  each  level  surface  of  nature  there  must  be  a  point 
or  points  of  maximum  or  minimum  potential.  A  continuous 
curved  line,  drawn  perpendicularly  to  each  of  the  level  surfaces 
which  it  intersects,  represents  a  line  of  action  or  tendency  to 
motion,  and  every  such  trajectory  must  finally  terminate  in  one  of 
the  included  points  of  maximum  or  minimum  potential.     Each  of 


—  33  — 

these  points  may  then  be  regarded  as  a  centre  of  action,  towards,  or 
from  which,  all  motion  tends  along  the  various  trajectories,  accord- 
ing as  the  point  is  that  of  a  maximum  or  a  minimum  potential. 

69.  If  the  'potential  has  a  constant  value  for  any  portion  of  space, 
this  same  constant  value  must  extend  throughout  all  that  space,  including 
this  portion,  for  ivhich  the  potential  and  all  its  derivatives  are  finite  and  con- 
tinuous functions.  For,  in  order  that  the  potential  may  be  absolutely 
constant  for  any  finite  extent,  however  small,  all  its  derivatives 
must  vanish.  But  it  follows,  from  Taylor's  Theorem,  that  the 
difference  of  the  value  of  the  potential  for  any  portion  of  space,  for 
wdiich  it  is  continuous  and  finite,  as  well  as  all  its  derivatives,  is  a 
linear  function  of  its  derivatives  at  any  point  of  that  space.  The 
difference  of  the  potential,  therefore,  vanishes,  when  all  the  deriva- 
tives vanish  and  the  potential  is  constant. 

The  portion  of  space,  for  which  the  derivatives  are  originally 
assumed  to  be  constant,  must  be  a  solid,  having  the  three  dimen- 
sions of  extension,  in  order  that  this  theorem  be  applicable. 

70.  Throughout  any  such  portion  of  space,  in  which  the 
potential  is  constant,  there  can  be  no  tendency  to  motion  in  any 
direction.  In  such  extent,  therefore,  there  can  be  no  mass  of 
matter,  for  it  is  contrary  to  experience  that  there  should  be  matter 
where  there  are  no  dynamical  phenomena. 

71.  In  all  the  observed  laws  of  material  action,  the  potential, 
which  belongs  to  the  action  of  each  particle  of  matter,  is  finite  and 
continuous,  as  well  as  all  its  derivatives,  for  the  whole  extent  of  space 
exterior  to  the  particle.  Hence,  the  potential  and  its  derivatives, 
for  every  system  of  nature,  are  finite  and  continuous  functions 
throughout  any  portion  of  space  which  contains  no  material  mass. 

72.  Hence,  it  follows,  that  for  every  finite  system  of  nature,  any 
portion  of  space,  in  ivhich  the  potential  is  constant,  must  he  finite,  and 
hounded  on  all  sides  hy  material  masses.     This  portion  of  space  cannot 


—  34  — 

extend  to  infinity,  because,  if  it  were  to  have  such  an  extent,  the 
finite  mass,  which  w^oukl  be  its  inner  limit,  would  exhibit  no 
external  indication  of  force  ;  whereas,  it  is  obvious  that  no  matter 
can  ever  have  been  observed,  except  by  such  a  manifestation  of  its 
existence. 

73.  There  are  forces  in  nature  which  are  temporarily  fixed,  and 
for  which  the  j)otential  may  vanish  throughout  all  space  exterior  to 
the  limit  in  which  the  centres  of  action  are  contained. 

74.  The  difference  between  the  values  of  the  potential  for  any 
two  points  may  be  computed  by  supposing  a  unit  of  mass  to  move 
from  one  point  to  the  other  upon  any  line  taken  at  pleasure,  and 
determining  the  change  of  power  which  it  receives  from  this 
motion.  The  change  of  the  potential  may  be  computed  for  each 
force  separately,  and,  in  making  the  partial  computations,  it  is 
sufficient  to  suppose  the  unit  of  mass  to  move  from  the  level 
surface  of  one  point  to  that  of  the  other,  and  one  of  the  perpen- 
dicular trajectories  may  be  taken  for  the  path  of  this  motion. 

75.  If,  in  any  system, 

F,  F',  &c.,  are  the  forces ; 

/,/',  &c.,  the  directions  in  which  they  act ;  and 
i2  is  the  value  of  the  potential ; 

the  general  expression  of  the  potential  for  any  point  of  the 
system  is 

a^^'/Fdf, 

in  which  the  limits  of  integration  extend  from  the  values  of/,/',  &c., 
w^hich  correspond  to  the  position  of  the  point,  to  infinity.  The 
expression  for  the  tendency  to  motion  in  any  direction,  as  that  of 
;?,  is 

D,ll  =  D,2'fFdf. 


II. 

COMPOSITION    AND    RESOLUTION    OF    FORCES. 

70.  No  phenomenon  is  observed,  in  which  a  single  force  acts 
freely  by  itself.  In  all  cases,  various  forces  are  combined  ;  and  it 
is  important,  therefore,  to  ascertain  what  are  the  dynamical  results 
of  such  combinations. 

77.  A  single  force  acts,  at  each  point,  perpendicularly  to  its 
level  surface,  with  an  intensity  which  is  measured  by  the  derivative 
of  the  potential,  taken  with  reference  to  the  element  of  direction  of 
the  force.  The  intensity  of  its  action,  in  any  other  direction,  is 
measured  by  the  derivative,  with  reference  to  the  element  of  that 
direction.  If  another  level  surface  is  drawn  infinitely  near  the  one 
which  passes  through  the  point,  the  action  in  any  direction  is 
inversely  proportional  to  the  length,  intercepted  by  the  surfaces, 
upon  a  straight  line  drawn  in  the  given  direction.  But  the  surfaces 
may,  for  this  purpose,  be  considered  as  reduced  to  their  parallel 
tangent  planes  at  the  given  point  •  and  the  length,  intercepted 
between  two  parallel  planes,  upon  a  straight  line,  is  proportional  to 
the  secant  of  the  angle  which  the  line  makes  with  the  perpen- 
dicular to  the  plane.  Hence,  the  action  of  a  force  in  the  direction 
of  any  line,  is  proportional  to  the  cosine  of  the  angle  which  it 
makes  with  the  direction  of  the  force. 

If,  then,  upon  a  straight  line  drawn  in  the  direction  of  a  force, 
a  length  is  taken  to  represent  the  intensity  of  the  force,  the  action 
in  any  direction  is  represented  by  the  projection  of  this  length 
upon  that  direction,  or  by  using  the  word  force  for  the  representa- 
tive of  the  force,  the  proposition  becomes,  that  the  action  of  a  force  in 
any  direction  is  the  projection  of  the  force  upon  that  direction. 

78.  When  several  forces  act  upon  a  point,  their  total  action  in 


—  3G  — 

any  direction  is  the  algebraic  sum  of  their  projections  upon  that 
direction. 

79.  When  three  forces,  ivhich  are  not  in  the  same  plane,  act  iq^on  a 
point,  their  combined  action  is  equivalent  to  that  of  a  single  force,  which  is 
represerded  in  magnitude  and  direction  ly  the  diagonal  of  the  parallelopiped 
constructed  upon  the  three  forces. 

For  the  algebraic  sum  of  the  projections  of  the  forces  upon  any 
direction  perpendicular  to  the  diagonal,  is  zero,  while  that  of  the 
projections  upon  the  diagonal  is  the  diagonal  itself 

80.  All  the  forces  ivhich  act  upon  a  point,  are  equivalent  to  a  single 
force,  which  is  called  their  resultant.     For  a  single  point  can  only  tend 

to  move,  with  a  certain  intensity,  in  some  one  direction,  however 
various  may  be  the  forces  which  act  upon  it ;  and  any  such 
tendency  to  motion  can  be  produced  by  one  force  acting  upon 
the  point. 

The  actions  of  all  the  forces  in  three  directions  which  are 
perpendicular  to  each  other,  can  be  found  by  §  78 ;  and  these  three 
partial  forces  can  then  be  combined  by  §  79  into  one  force  which 
will  be  the  resultant.  But  the  following  method  of  finding  the 
resultant  illustrates  the  use  which  may  be  made  of  the  level 
surfaces. 

81.  In  considering  the  action  of  a  force  upon  a  fixed  point  in 
space,  the  variable  character  of  the  force  for  other  points  of  space 
may  be  neglected,  and  its  level  surfaces  may  be  regarded  as  parallel 
planes  perpendicular  to  the  direction  of  the  force.  Thus,  it  may  be 
assumed  that 

Ff  is  the  potential  of  the  force  F,  which  acts  in  the  direction 
:  of/;  for 

Df{Ff)  =  F,  is  the  intensity  of  the  force  ;  and 
Ff=z  a  constant,  or 
/  =  a  constant, 


—   o<    — 

is  the  equation  of  a  plane  perpendicular  to  /.     Hence,  the  potential 
of  all  the  forces  which  act  upon  the  point,  is 

If  then 

Pq  is  the  resulting  force  resolved  in  the  direction  of  q ;  if 
p  is  the  direction  of  the  resultant,  and 
P  is  the  resultant ; 

the  value  of  either  of  these  forces  is  represented  by  the  formula 

P^  =  D^n  =  ^'FDJ=  ^'Fcosf. 
But,  by  putting 

the  condition  that  ]?  is  perpendicular  to  the  level  surface,  for  which 
the  potential  is  constant,  gives 


X           L 

=^ 

p. 

L' 

Hence  the  value  of  the  resultant  is 

P 

=  i>,i2=.^.. 

D. 

.^D, 

,x 

==-^,i>,i2cos 

\P  ■ 

X 

—  ^ 

PI 

L 

~      L     ~  L 

T  J  (  ^ 

p5 

!\ 

82.  By  an  elementary  motion  of  translation,  each  point  of  a 
system  is  carried  to  the  same  distance  in  the  same  direction ;  the 
potential  of  the  system  is  changed,  therefore,  precisely  as  if  all  its 
points  were  united  in  one,  and  all  the  forces  applied  at  this  point. 
The  tendency  of  a  system  to  any  motion  of  translation,  is,  then,  the  same  as 


—  38  — 

that   ivhich  imild  arise  from  the   action   of  a  single  force,  equal  to  the 
resultant  of  all  the  forces,  supposed  to  he  applied  at  the  same  point. 

83.  The  moment  of  a  force,  ivith  reference  to  a  point,  is  the  product 
of  the  force  multiplied  by  its  distance  from  the  point.  The  moment 
of  a  force,  with  reference  to  a  line,  is  the  product  of  the  projection  of 
the  force  upon  a  plane  perpendicular  to  the  line  multiplied  by  the 
distance  of  the  force  from  the  line. 

The  moment  of  a  force,  with  reference  to  a  line,  may  be 
rej)resented  geometrically  by  a  corresponding  length  taken  upon 
the  line,  and  the  name  of  the  moment  may  be  given  to  its  geomet- 
rical representative. 

The  moment  of  a  force,  ^Yith  reference  to  a  point,  is  the  same 
with  the  moment,  with  reference  to  the  line,  which  is  drawn 
through  the  point  perpendicular  to  the  common  plane  of  the  point 
and  the  force. 

84.  The  moment  of  a  force,  ivith  reference  to  a  line  passing  through  a 
point,  is  equal  to  the  projection  upon  the  line  of  the  moment,  tvith  reference  to 
the  point.  For  the  moment,  with  reference  to  the  point,  is  equal  to 
double  the  area  of  the  triangle,  of  w^hich  the  base  is  the  force,  and 
the  altitude  is  the  distance  of  the  force  from  the  point ;  and  the 
moment,  with  reference  to  the  line,  is  equal  to  double  the  area  of 
the  triangle,  of  which  the  base  is  the  projection  of  the  force  upon 
the  plane  perpendicular  to  the  line,  and  the  altitude  is  the  distance 
of  this  projection  from  the  line.  But  the  latter  of  these  triangles  is 
the  projection  of  the  former  upon  the  plane,  and  its  area  is  equal  to 
the  product  of  the  area  of  the  former  triangle,  multiplied  by  the 
cosine  of  the  angle  of  the  planes  of  the  two  triangles.  But  the 
lines  upon  Avhich  the  moments  are  represented,  being  resj)ectively 
perpendicular  to  these  planes,  have  the  same  mutual  inclination. 
The  moment,  with  reference  to  the  line,  is,  therefore,  equal  to  the 
product  of  the  moment,  with  reference  to  the  point,  multiplied  by 


—  39  — 

the  cosine  of  the  mutual  angle  of  the  moments ;  that  is,  it  is  equal 
to  the  iDrojection  upon  the  line  of  the  moment,  with  reference  to 
the  point. 

85.  Hence  it  follows  that  the  moments  of  forces,  with  refer- 
ence to  points,  may  be  combined  by  the  same  processes  in  Avhich 
the  forces  themselves  are  combined,  and  that  all  the  moments,  ivith 
reference  to  a  jwint,  may  he  combined  into  one  resultant  moment. 

86.  The  tendency  of  the  force  F,  of  which  the  potential  is 
Ff,  to  produce  an  elementary  rotation,  di),  about  a  line  p,  is 


But  if 

(1426)  gives 


Q  is  the  distance  of  F  from  ^7, 


A/=(>sin^; 
the  projection  of  jP  upon  the  plane  perpendicular  to^j,  being 

i^sin-^, 

the  tendency  to  rotation  about/?  becomes 

Q  Fmi^l.  =  the  moment  of  F  w^ith  reference  to  j) ; 

that  is,  the  moment  of  a  force,  tvith  reference  to  a  line,  is  the  measure  of  its 
tendency  to  ^produce  rotation  ahout  that  line. 

87.  The  direction  of  the  positive  moment  must  be  assumed  to 
be  the  same  with  that  of  the  axis,  about  which  the  tendency  to 
rotation  of  the  force  is  positive. 

88.  The  resultant  moment  of  all  the  forces  of  a  system,  loitli  reference 
to  a  point,  is  the  measure  of  their  tendency  to  produce  rotation  ahout  that 
point.  Hence,  the  one  force,  of  which  the  moment  is  equal  to  the 
resultant  moment,  has  the  same  tendency  to  produce  rotation. 


.         —  40  — 

89.  The  resultant  moment  of  all  the  forces  which  act  upon  a 
point,  with  reference  to  any  line  or  to  any  other  point,  is  the  same 
with  the  moment  of  their  resultant.  For  the  point  upon  which  the 
forces  act  tends  to  move  in  the  direction  of  their  resultant,  with  a 
force  equal  to  its  intensity,  and  its  moment  is,  therefore,  the 
measure  of  the  tendency  to  motion. 

90.  The  moment  of  a  force,  with  reference  to  a  line  y,  is 
equal  to  its  moment,  with  reference  to  a  parallel  line  />,  increased 
by  the  moment  of  an  equal  and  parallel  force,  acting  at  any  point 
of  the  line  p.  For  the  distance  of  the  original  force  from  the  line 
y,  is  equal  to  its  distance  from  the  line  p,  increased  by  the  distance 
from^'  of  the  parallel  force  passing  through  jk>. 

91.  Hence  the  resultant  moment  of  any  forces,  with  reference  to  a  line 
p,  is  equal  to  their  resultant  moment,  ivith  reference  to  a  parallel  line  p, 
increased  hy  the  moment,  with  reference  to  p,  of  equal  and  parallel  forces 
acting  at  any  point  of  the  line  p. 

92.  The  residtant  moment  of  any  forces,  ivith  reference  to  a  point  0' , 
is  equal  to  their  residtant,  ivith  reference  to  a  ^^oint  0,  increased  by  the 
moment,  ivith  reference  to  0' ,  of  equal  and  parallel  forces  acting  at  0.  For 
this  proposition  is  true  for  each  pair  of  the  parallel  axes  of  two 
parallel  systems  of  three  rectangular  axes,  of  which  the  points  0 
and  0'  are  the  respective  origins. 

93.  A  coujyle  of  forces  is  a  system  of  two  parallel  and  equal 
forces  which  act  in  different  lines. 

94.  The  moment  of  a  couple  of  forces  has,  for  every  point  of  space, 
the  same  value,  ivhich  is  equal  to  the  moment  of  one  of  them  for  any  point  of 
the  other.  For  two  forces,  equal  and  parallel  to  them,  applied  at  any 
point,  destroy  each  other's  action,  and  their  resultant  vanishes. 

95.  The  tendency  of  a  couple  of  forces  to  produce  rotation 
about  a  point,  is  the  same  as  that  of  any  system  of  forces,  when  its 
moment   is    equal   to    the    resultant   moment   of  the  system,  with 


—  41  — 

reference  to  the  point.  But  the  couple  has  no  tendency  to 
produce  a  transhxtion ;  whereas  the  resultant  of  a  system  of  equal 
and  parallel  forces,  acting  at  the  point,  has  all  the  tendency  of  the 
system  to  produce  translation,  but  none  to  produce  rotation  about 
the  point.  Hence,  the  three  forces,  of  zuhich  one  is  the  resultant  of  the 
equal  and  parallel  forces  acting  at  a  point,  and  the  other  two  constitute  a 
couple,  of  2uhich  the  moment  is  the  same  zuith  the  resultant  moment,  ivith 
reference  to  the  point,  f idly  represent  any  system  of  forces  in  their  tendency 
to  produce  rotation  and  translation. 

96.  Since  the  position  of  the  couple  of  forces  is  quite  arbi- 
trary, one  of  the  pair  may  be  taken  to  act  at  the  same  point  with 
the  resultant  of  all  the  forces;  and,  by  combining  it  with  the 
resultant,  the  system  of  three  forces  may  be  reduced  to  two. 

97.  A  point  can  always  be  found  in  space,  for  which  the 
moment  of  a  given  force  has  any  assumed  magnitude,  and  any 
direction  which  is  perpendicular  to  the  force.  Because  the  distance 
of  the  point  from  the  force,  which  is  one  of  the  factors  of  the 
moment,  may  vary  from  zero  to  infinity,  and  its  direction  from  the 
force  may  be  that  of  any  perpendicular  to  the  force. 

Hence,  if  the  resultant  moment,  with  reference  to  a  point  0, 
of  any  system  of  forces,  is  decomposed  into  two  moments,  of  which 
one  has  the  same  direction  with  the  force,  and  the  other  is  per- 
pendicular to  it,  another  point  0'  can  be  found,  for  which  the 
moment  of  the  resultant,  acting  at  0,  is,  in  amount  and  direction, 
the  negative  of  that  component  of  the  resultant  moment  for  0, 
which  is  perpendicular  to  the  resultant.  For  the  point  0' ,  there- 
fore, the  resultant  moment,  coincides  in  direction  with  the  result- 
ant itself;  and  of  the  three  corresponding  forces  which  represent 
the  tendency  of  the  system  to  produce  rotation  and  translation,  the 
plane  of  the  couple  is  perpendicular  to  the  direction  of  the  result- 
ant. 

6 


—  42  — 

98.  If  all  the  forces  lie  in  the  same  plane,  for  any  point  of  the 
plane  the  moment  of  each  of  the  forces  is  perpendicular  to  the 
plane,  and,  therefore,  the  resultant  moment  is  perpendicular  to  the 
plane.  But  the  resultant  of  the  parallel  and  equal  forces  acting  at 
the  point  must,  if  it  does  not  vanish,  lie  in  the  same  plane,  and  be 
perpendicular  to  the  resultant  moment.  If,  then,  the  resultant  does 
not  vanish,  a  point  of  the  plane  can  be  found  for  which  the  result- 
ant moment  vanishes. 

99.  If  all  the  forces  are  parallel,  the  moment  of  each  of  them, 
for  any  point,  lies  in  the  plane  which  is  drawn  through  the  point 
perpendicular  to  the  forces.  But  the  resultant  of  the  parallel  and 
equal  forces,  acting  at  the  point,  has  the  same  common  direction 
with  them,  and  is,  therefore,  perpendicular  to  the  resultant  moment. 
If,  then,  the  resultant  does  not  vanish,  a  point  can  be  found  for 
w^hich  the  resultant  moment  vanishes. 

Hence,  if  all  the  forces  of  a  sf/stem  lie  in  ilie  same  plane,  or  if  they  are 
all  parallel  to  each  other,  their  tendency  to  produce  translation  or  rotation  is 
eqidvalent,  either  to  that  of  a  single  force,  or  to  that  of  a  couple  of  forces. 

100.  If  of  any  system  of  forces,  and  for  a  point  0 

J/ is  the  resultant  moment, 

R  the  resultant  of  equal  and  parallel  forces  acting  at  0, 
Mp  and  R^  the  projections  of  J/ and  R  upon  the  direction 
of  77, 

and  if  the  same  letters  accented  denote  the  same  quantities  for  the 
point  0',  and  if 

X,  y,  and  z  are  the  rectangular  coordinates  of  0'  with  reference 
to  0, 

the  value  of  the  moment   of  the   forces  for   either   of  the   axes 
passing  through  0'  is, 

M'^^M^  —  zR^-^^yR^. 


—  43  — 

But  if  the  direction  of  the  axis  of  z  is  assumed  to  be  the  same  with 
that  of  R,  these  moments  become 

3C=3I,. 

The  coordinates  of  the  points,  for  which  the  resultant  moment  has 
the  same  direction  with  tlie  resultant,  are 

71/,         My^ 

^"~        'R'^—  R' 

101.  The  number  of  forces  which  is  required  to  produce  any 
of  the  special  effects  of  a  given  system  of  forces,  is  usually  much 
less  than  the  whole  number  of  those  which  actually  concur  in 
their  production.  The  mode  of  analj'sis,  by  which  the  requisite 
forces  may  be  ascertained,  is,  in  most  cases,  quite  as  simple  as  that 
by  which  the  effects  of  rotation  and  translation  have  been  investi- 
gated. 

III. 

GRAVITATION,    AND    THE    FORCE    OF    STATICAL    ELECTRICITY. 

102.  Gravitation  is,  among  all  the  forces  of  nature,  conspicuous 
for  its  universality,  and  the  grandeur  of  the  scale  upon  which  it  is 
exhibited. 

Each  particle  of  matter  is  an  elementary  centre  of  action  for  the  force 
of  gravitation,  and  all  the  level  surfaces  for  each  imrticU  are  spherical 
surfaces,  of  which  the  particle  is  the  centre.  The  value  of  the  potential  for 
any  particle,  is  inversely  proportional  to  the  distance  from  the  particle,  and 
for  different  particles  it  is  proportional  to  the  mass  of  the  particle. 

103.  Another  force  which  seems  to  be  equally  universal  with 
gravitation,    and   of  which   gravitation   has   been,   perhaps  justly, 


—  44  — 

regarded  as  a  residual  force,  and  which  is  subject  to  the  same  law, 
in  respect  to  distance  from  each  elementary  centre  of  action,  is  that 
of  statical  electricity.  This  force,  however,  is  endowed  with  duality, 
and  consists  of  tivo  forces,  of  ivhicli  one  has  a  positive,  and  the  other  a 
negative  potential.  Both  forces  are  usually  combined  with  equal 
intensity,  in  the  same  centre  of  action,  so  as  to  neutralize  each 
other's  influence,  and  thus  lie  dormant.  With  each  of  these  the  poten- 
tial is  positive  in  reference  to  electricity  of  the  other  Jcind,  and  negative  iiith 
reference  to  that  of  the  same  Jcind.  The  tendency  to  motion,  arising 
from  one  kind  of  electricity,  is  exactly  equal  and  opposite,  then,  to 
that  which  arises  from  the  action  of  an  equal  intensity  of  the  other 
kind,  distributed  in  the  same  way. 

104.  The  action  of  electricity  upon  the  mass  of  a  particle 
is  indirect ;  the  direct  action  is  upon  the  electricity  associated 
with  the  mass.  In  most  bodies  the  electricity  yields  with  more  or 
less  facility  to  this  action,  leaves  the  particle  with  which  it  is 
originally  combined  for  another  particle,  and  finally  assumes  such  a 

form  of  distribution  within  and  upon  the  hodg,  tJiat  the  tendency  to  motion 
shall  nowhere  exceed  the  resistance  to  motion.  Bodies  in  which  there  is 
no  resistance  to  the  motion  of  electricity  are  called  perfect  conductors; 
while  those  in  which  the  resistance  is  infinite  are  called  perfect  non- 
conductors. 

105.  Let 

dm  denote  the  mass  of  a  particle  of  matter  in  the  case  of 
gravitation,  or  the  value  of  its  potential  at  the  unit  of 
distance,  in  the  case  either  of  gravitation  or  elec- 
tricity ; 

da,  the  element  of  volume  of  the  mass  ; 

Jc,  the  density  of  the  matter,  in  the  case  of  gravitation,  or 
the  intensity  of  the  force  of  electricity,  compared 
with  the  unit  of  intensity  ; 


—  45  — 

/,  the  distance  from  the  particle  ; 
cUl,  the  value  of  the  potential  for  the  particle  ; 

the  expression  of  the  potential  for  the  particle  is 

,  ^  dm         hda 

The  general  value  of  the  potential  for  the  whole  body  is 

106.     With  reference  to  a  system  of  three  rectangular  axes, 
let 

X,  I/,  z,  be  the  coordinates  of  the  point  in  space,  for  which  the 

potential  is  12,  and 
^,  7j,  i',  those  of  the  particle. 

Adopt  also  the  functional  notation 
The  derivatives  of/ and /~^  are 

A/=COSy  =  -— r-. 


j^,f^P-(—^^_^ 


Hence 


r  f 

—  l-f  3cos%^ 

—  J.  . 

1        _3_l_32;cos2- 


—  46  — 
and,  therefore, 

This  last  equation,  which  is  called  Laplace's  equation,  only 
applies  to  that  extent  of  space  for  which  the  derivatives  of  the 
jDotential  are  continuous  functions,  that  is,  where  there  are  no 
centres  of  action ;  but,  where  there  are  centres  of  action,  it  requires 
a  modification  which  will  soon  be  investigated.  The  integration  of 
this  equation,  combined  with  peculiar  considerations  in  special 
cases,  gives  the  value  of  the  potential  for  all  the  problems  of 
gravitation  or  statical  electricity. 

107.  T/ie  tendency  to  motion,  resulting  from  the  gravitating  or 
electrical  action  of  a  particle  of  matter,  heing  normal  to  the  level  surface,  is 
directed  in  the  straight  line  draimi  to  the  imrticle.  Its  intensity  is  the 
derivative  of  the  potential,  and  expressed  by  the  equation. 

The  force  of  the  gravitating  or  electrical  action  of  a  particle  of  matter, 
is,  therefore,  inversely  froportional  to  the  srpiare  of  the  distance  from  the 
particle.  It  is  attraction  in  the  case  of  gravitation,  or  hetween  electricities 
of  opposite  lands,  and  repidsion  hctiuccn  electricities  of  the  same  Jdnd. 

ATTRACTION    OF    AN    INFINITE     LAMINA. 

108.  The  investigation  of  the  potential  of  a  lamina  of  uniform 
density,  and  included  between  two  infinitely  extended  planes,  is 
simplified  by  the  consideration,  that  it  must  have  the  same  value 
for  all  points  of  space  which  are  at  the  same  distance  from  either 
surfiice  of  the  lamina.  Because  all  such  points  are  similarly  situ- 
ated with  reference  to  the  lamina,  on  account  of  its  infinite  extent. 
Hence,  if  either  surface  of  the  lamina  is  adopted  for  the  plane  of  g  z, 


—  47  — 

the  derivatives  of  the   potential,  with  reference  cither  to  y  or  z 
must  vanish,  and  Laplace's  equation  becomes 

The  integral  of  this  equation  gives  the  value  of  the  potential, 
for  a  point  external  to  the  lamina,  or  upon  its  surface, 

in  which  A  and  B  are  arbitrary  constants. 

109.  The  level  surfaces  are  the  planes  determined  by  the 
equation  (477),  when  il  is  the  constant  value  of  the  potential  for 
the  level  surface. 

110.  The  action  of  the  lamina  upon  any  external  point,  is  in 
a  direction  perpendicular  to  either  surface,  and  its  force  of  attraction 
or  repuhion  is  constant  upon  all  points,  for  it  is  given  by  the  equation 

I),S2=A. 

111.  The  values  of  A  and  B  in  any  special  case  must  be 
ascertained  by  direct  integration.  The  integration  indicated  in 
(458),  gives  an  infinite  value  of  the  potential,  whereas  the  integra- 
tion of  its  derivative,  with  reference  to  x,  gives  A  itself,  in  a  finite 
form,  which  shows  that  the  infinite  portion  of  the  potential  belongs 
to  B.  The  integration  for  finding  the  derivative  of  the  potential  is 
effected  by  putting 

Q=ffimJ, 

=  the  projection  of/  upon  the  plane  of  ?/:s. 
a  =  the  thickness  of  the  lamina ; 
whence 

f=^{x  —  ^)secj, 

Q  =  [x  —  ^')tan'^. 


—  48  — 


da  =  Qclod^d^, 


^■cosj 

7^ 


=-J-- i  J?"^"^- 

0         0         0 

=  —2jia/c  =  A. 


This  value  of  A  corresponds  to  a  positive  value  of  .r,  but  for  a  nega- 
tive value  of  X  its  sio:n  must  be  reversed. 

112.  For  a  point  situated  within  the  lamina,  a  plane  may  be 
drawn  through  it  parallel  to  the  superficial  planes,  and  dividing  the 
lamina  into  two  partial  laminae,  of  which  the  thicknesses  are  a:  and 
a  —  X.     Hence,  the  value  of  the  derivative  of  the  potential  is 

DJ2  =  —  27i/cx-{~2Tt/c{a  —  x) 

=.2nk{a  —  2x). 

poisson's  modification  of  Laplace's  equation  for  an  interior  point. 

113.  The  modification  which  is  required  of  Laplace's  equa- 
tion, in  order  that  it  may  be  apj)licable  to  any  point  of  an  acting 
mass,  must  be  the  same  for  all  cases.  For  it  would  not  be  needed, 
if  the  point  of  action  were  contained  within  any  extent,  however 
small,  of  void  space.  It  depends,  therefore,  exclusively  upon  the 
infinitesimal  portion  of  matter  at  the  point,  and  is  unaffected  by 
any  variations  in  the  form  and  extent  of  the  acting  body.  It  need 
be  investigated,  then,  in  only  a  single  case.  Now  the  derivative 
of  (48i6)  gives 

which  substituted  in  Laplace's  equation  gives  for  an  internal  point 


—  49  — 
of  the  infinite  lamina, 

wliicli  is,  therefore,  the  required  modification  of  this  equation. 
This  modified  equation,  in  which  Jcq,  denotes  tlie  value  of  h  at  the 
point  of  action,  is  applicable,  as  remarked  by  Sturm,  even  when  the 
point  is  exterior  to  the  body.  This  same  geometer  has  observed 
that,  by  supposing  the  value  of  Jc  gradually  to  shade  off  from  its 
value  within  the  body  to  zero,  this  graduation  occurring  within  an 
infinitely  small  extent,  so  as  not  sensibly  to  interfere  w^th  the 
actual  phenomena  of  nature,  the  potential  and  it^  differential  coeffi- 
cients may  become  continuous  functions.  It  must  be  further 
observed,  however,  tliat  this  imaginarj^  graduation  must  extend 
throughout  all  space,  although  Jc  must  have  an  infinitesimal  value 
where  there  is  no  portion  of  active  force  ;  for  if  it  were  to  vanish 
throughout  any  finite  portion  of  space,  however  small,  the  reason- 
ing of  §  69,  would  prove  that  all  the  derivatives  of  the  potential 
were  not  finite  and  continuous. 

ATTRACTION    OF    AN   INFINITE    CYLINDEE. 

114.  The  investigation  of  the  potential  of  an  infinite  cylinder 
is  simplified  by  the  consideration  that  its  value  must  be  the  same 
for  all  points  situated  upon  the  same  straight  line  parallel  to  one  of 
the  sides  of  the  cylinder.  If  this  direction  is  adopted  for  the  axis 
of  s,  the  derivative  of  the  potential,  with  reference  to  .e,  must 
vanish,  and  Laplace's  equation  becomes 

The  integral  of  this  equation  is 

7 


—  50  — 

in  which  cF  and  S^j  are  arbitrary  functions,  and  must  be  determined 
for  each  case  by  special  considerations. 

115.  The  level  surfaces  are  the  cylindrical  surfaces,  of  which 
(49go)  is  the  general  equation,  if  i2  has  the  constant  value  belonging 
to  that  surface. 

116.  The  attraction  in  the  direction  of  the  axis  of  x  is 

in  which  the  accents  denote  the  derivatives  of  the  functions,  with 
reference  to  their  explicit  variables. 

The  attraction^n  the  direction  of  the  axis  of^  is 

D^ii  =  [W'i^  +  i/vC^)  -  g!((^  -yv'~.)]vC"i. 

The  whole  action  is,  then, 

V/[(A)^  +  (i>,)^]-<2  =  2v/[9^'(^ +^V^).3^;(^-y\C^)]. 

117.  When  the  jwint  of  action  is  so  far  from  the  cijllnder  that  the 
square  of  the  linear  dimensions  of  the  hase  can  le  neglected,  in  comparison 
ivith  the  square  of  the  least  distance  of  the  point  from  the  cylinder,  the 
problem  can  be  greatly  simplified. 

Find  in  this  case  a  line  parallel  to  the  axis  of  z,  of  which  the 
co'drdinates  a  and  h,  with  reference  to  the  axes  of  x  and  y,  are 
determined  by  the  equations 


am. 


O  in  O  m 

J{V — Z*)  =  0  =  /   1]  —  Im. 
in  U  m 


This  line  may  be  called  the  axis  of  gravity  of  the  cylinder,  and 
its  position  is  wholly  independent  of  the  directions  of  the  axes  of 
90  and  y.     For  the  conditions  by  which  this  axis  is  determined  will 


—  51  — 

give,  with  regard  to  any  other  axis  of  x,  with  reference  to  which 
the  notation  is  distinguished  by  the  subjacent  numbers, 

r(i'i— «i)=r(^— «)cos^ +r  (j/— ^)cos^ =o. 

If  the  axis  of  gravity  is,  then,  assumed  for  the  axis  of  s,  the 
equations  (5025_26)  become 

or 

118.  Since,  from  the  nature  of  the  cylinder,  the  functions 
which  are  here  to  be  integrated  are  independent  of  t,  these 
equations  give 

119.  Let  the  perpendicular  from  the  point  of  action  upon  the 
axis  of  2^  be  assumed  for  the  axis  of  x,  and  let 

/q  be  the  distance  of  the  point  of  action  from  the  projection  of 

any  particle  of  the  cylinder  upon  the  axis  of  z, 
Q  the  distance  of  the  particle  from  the  axis  of  ^. 

The  conditions  of  the  problem  under  consideration  give 

/^  =  (.r  -  |)=  +  7;=  +  f  ^  =/5  _  2  .T?  +  </, 

/  Jo  ^  Jo/  Jo  Jo 

J  ■mj  J  mJo         J  mj  q 

J  m  So         J  C/u  J  ^J  V  J  mfo 


f^9 


SO  that  the  iwtential  is  the  same  as  if  all  the  ^yarticles  of  the  cylinder  were 
united  in  their  projections  upon  the  axis  of  gravity,  tvhen  the  point  is  at  a 
sufficiently  great  distance  from  the  cylinder. 

120.  By  letting 

K  denote  the  mtensity  of  the  action  concentrated  upon 
each  point  of  the  axis  of  gravity  when  the  cylinder 
is  projected  npon  it ; 

the  value  of  the  whole  action  of  this  axis  is 

D,n  =  —  f^=  —Kx  f.  ,_[.,,§ 

00  00 

= l.^cos';  = , 

or  the  potential  is 

£2  =  —  2mogx  +  B, 

in  -which  the  arbitrary  constant  B  is  infinite. 

121.  When  the  base  of  the  cylinder  is  the  space  zvhich  is  contained 
hetween  two  concentnc  circles,  the  axis  of  gravity  coincides  with  the 
geometrical  axis,  the  potential  is,  from  the  symmetry  of  the  figure, 
the  same  in  all  directions  from  the  axis,  and  its  value  only  depends 
upon  the  distance  from  the  axis.  Let  the  axes  be  the  same  as  in 
§§117  and  119,  except  that  the  point  of  action  is  in  the  plane  of 
X  y,  but  not  in  the  axis  of  x,  and  let 

r  =.  the  radius  vector  of  the  point  of  action,  and 
e  =  the  base  of  the  Naperian  system  of  logarithms. 

The  potential  is  a  function  of  r,  and  does  not  involve  the  inclination 
of  r  to  the  axis  of  x.     Hence 


—  53  — 

But  by  (493o) 


whence 


and 


Dr^n  =  \7)Arc       )rc       —^■''\rc         )rc         \sJZi=^', 


But  the  two  members  of  this  equation  are  functions  of  two  different 
and  independent  variables,  which  are 

re         and  re  ' 

and,  therefore,  neither  can  be  contained  in  the  vahie  of  the  other, 
so  that  each  of  them  disappears  from  their  common  value,  which  is, 
therefore,  constant.  With  regard  to  any  variable  whatever,  there- 
fore, this  equation  gives 


rWrz=rW-yr^A, 
and,  by  integration. 


r 


^^r=^A\ogr-\-B^ 
?Fir  =  ^logr4-i?i. 

The  value  of  the  potential  is,  then,  if  the  two  constants  are  com- 
bined in  one, 

i2=^log(r/  "j  +  AlogKre  '      )+B.^ 
=  2^1ogr  +  i?2, 

and   the   action  upon  the  point  is  in  the  direction  of  r,  and   its 


—  54  — 
value  is 

r 

122.  When  the  point  of  action  is  upon  the  axis,  it  is  plain, 
from  the  synnnetrical  nature  of  the  cylinder,  that  the  action  is 
cancelled  in  each  direction,  and  in  this  case 

Ai2  =  ?^  =  0, 

whence 

^  =  0. 

For  every  point  ivitJiin  the  inner  cylindrical  howidary  of  tJiis  cylindrical 
shell,  the  action,  therefore,  vanishes,  and  the  potential  is  constant. 

123.  When  the  point  of  action  is  without  the  cylinder,  the 
constants  are  found  by  the  condition  that  when  the  distance  is  very 
great,  the  value  must  be  the  same  as  that  of  (52i3).     Hence 

A  =  —  K, 

that  is,  the  action  upon  every  point,  tvithoiit  the  circular  cylinder,  is  the 
same  as  if  the  ivhole  mass  of  the  cylinder  ivere  concentrated  icpon  its  axis. 

124.  No  other  case  of  the  infinite  cylinder  is  of  sufficient 
interest  to  divert  the  current  of  the  work  from  the  finite  masses 
of  nature. 

RELATION    OF    THE    POTENTIAL    TO    ITS    PARAMETER. 

125.  The  varying  value  of  the  potential  from  one  level 
surface  to  another,  depends  upon  the  law  of  the  change  of  surface, 
and  may  be  represented  as  a  function  of  a  variable,  which  may  be 
called  its  ^^a^ccmeter.     Let 

I  denote  the  parameter  of  the  potential,  and  adopt  the  func- 
tional notation 


—  55  — 
The  derivative  of  the  potential  gives 

m  =z  Dinni  +  Djin = —4.7tk„ 

which  is  a  transformation  given  by  Lame. 

126.     For  a  point  of  void  space,  this  equation  gives 

by  which  the  potential  may  be  determined  for  given  forms  of  \ 


ATTRACTIOX    OF   A   FINITE   POINT   UPON   A   DISTANT   MASS.      CENTRE    OF    GRAVITY. 

127.  In  every  finite  mass  there  is  a  point  called  the  centre 
of  gravity,  of  which  the  coordinates  are  determined  by  equations,  for 
each  axis,  which  are  similar  to  (5025_26)-  This  point  is  independent 
of  the  positions  of  the  axes,  for  these  equations  give  for  any  other 
axis 

X  (I,  -  « ,)  =  ^«  (X,  (5  -  «)  <:°«".)  =  0  • 

If  the  centre  of  gravity  is  adopted  for  the  origin  of  coordinates, 
these  equations  are  reduced  to  (5l8_io). 

128.  "VYhen  the  point  of  action  is  so  far  from  the  attracting 
mass,  that  the  squares  of  the  linear  dimensions  of  the  mass  may  be 
neglected  in  comparison  with  the  square  of  the  distance  of  the 
point  from  the  mass,  the  formula  becomes 

^,,2_|_^2_^^(2.rs^)=r^-^.(2.t:0 

%J  mj  U  m  ^'  I  ' 


—  56  — 


that  is,  the  potential  of  a  fnite  point,  for  a  mass  tvJiich  is  so  remote  that 
the  square  of  the  linear  dimensions  of  the  hodf/  may  he  neglected,  in  compari- 
son tvith  the  square  of  the  distance  of  the  point  from  the  hody,  is  the  same  as 
if  the  body  tvere  concentrated  at  its  centre  of  gravity. 

ATTRACTION  OF  A  SPHEKICAL  SHELL. 

129.  In  the  case  of  a  shell  of  homogeneous  matter,  contahied 
between  the  surfaces  of  two  concentric  splieres,  the  value  of  the 
potential  must,  from  the  symmetry  of  the  figure,  depend  exclu- 
sively upon  the  distance  from  the  centre ;  and  for  the  same  reason 
this  centre  is  the  centre  of  gravit3^  If  the  centre  is  adopted  for  the 
origin  of  coordinates,  the  parameter  may  be  assumed  to  be  the 
radius  vector,  or  any  function  of  it.     Patting,  then, 

derivation  gives 

ni  =  4:2,x''  =  ir^ ^  il, 

Hence,  (S-Sg)  becomes 

A.(logi),i2)  =  -l. 

The  integral  of  this  equation  is,  by  the  introduction  of  the  arbitrary 
constants  A  and  B, 

n^B  —  4,  =  B  —  -. 


—  57  — 

130.  When  the  point  of  action  is  at  the  origin,  the  value  of 
the  potential  is  easily  obtained  by  direct  integration.  Let  in  this 
case 

^)o  and  ^1  be  the  internal  and  external  radii  of  the  spherical 
shell, 

iiiq  and  nii  the  masses  of  two  homogeneous  spheres  of  the  same 
density  with  the  shell,  and  of  which  the  radii  are  respect- 
ively ()o  and  (jj ;  and 

d\\i  the  elementary  solid  angle  of  which  the  vertex  is  at  the 
point  of  action. 

The  mass  of  the  shell  is 

m  =  m  I  —  ;^?  Q  =  4  7T  /"  (o  \  —  {tl)^ 
and  the  element  of  mass 

The  value  of  the  potential  is,  therefore, 

=  y^"  f  ((>i  —  (,>o)  =  ^nlciiil  —  (>l) 

^V^i  Go/ 

131.  ^YJlcn  the  imnt  of  action  is  in  the  interior  void  space  of  the"- 
shell,  the  constants  of  (5630)  must  have  the  same  values  as  at  the 
origin,  where  r  vanishes.     Hence,  for  this  space,  the  constants  are 

^  =  0, 

The  value  of  the  potential  in  the  interior  void  space   is,   therefore, 
constant,  and  there  is  no  tendency  to  motion  in  any  direction. 

8 


—  58  — 

132.  For  an  exterior  imnt,  the  j^otential  vanishes  when  r  is 
infinite,  while  for  a  point  at  a  great  distance  from  the  origin,  its 
value  is,  hy  §  128,  the  same  as  if  the  Avhole  mass  were  concentrated 
at  the  origin.     The  value  of  the  constants  in  this  case  are  then 

i?  =  0, 

yl  =  —  m\ 
and  the  potential  is 

r 

Any  exterior  iwint  is,  then,  attracted  ly  a  homogeneous  spherical  shell, 
precisely  as  if  the  luhole  mass  of  the  shell  tvere  concentrated  upon  its  centre 
of  gravity. 

ACTION    AND    REACTION    OF    A    SURFACE     OR    INFINITELY    THIN    SHELL 
OF     FINITE     EXTENT.       CHASLESIAN    SHELL. 

133.  An  infinitely  thin  shell  may  be  reduced  to  either  of  its 
surfaces,  upon  which  all  its  acting  force  may  be  concentrated,  and 
the  intensity  of  the  action  at  each  point  of  the  surface  will  be  the 
product  of  the  corresponding  intensity  of  the  force  of  the  shell, 
multiplied  by  the  thickness  of  the  shell,  and  the  element  of  the 
surface  must  be  substituted  for  the  element  of  volume  of  the  shell. 
Let  then, 

do  be  the  element  of  the  surface, 
*        N  the  exterior  direction  of  the  normal  to  the  surface, 

Jc  the  concentrated  intensity  of  action  at   any  point   of  the 

surface, 
d\\i  the  elementary  solid  angle  subtended  by  the  element  of  the 

surface  at  the  point  of  action  ; 

the  expression  of  the  element  of  the  surface  is 


do  =f^diiiseG^. 


—  59  — 

Hence 

—  Ic  dw  = ^k  da. 

The  second  member  of  this  equation  denotes  the  action 
exerted  bj  each  element  of  the  surface  in  a  direction  normal 
to  the  surface,  and  towards  the  interior  of  the  surface.  If,  there- 
fore, the  intensity  of  action  is  constant  over  the  surface,  the  action 
normal  to  the  sm^face  is  proportional  for  each  element  of  the 
surface,  to  the  solid  angle  subtended  by  the  element,  and  the  toted 
amount  of  the  action^  normal  to  the  surface,  exerted  hj  any  continuous  extent 
of  the  surface,  is  jproportional  to  the  v:hole  solid  angle  sultcnded  hj  the 
houndary  of  the  surface. 

134.  If  the  surface  is  a  plane,  the  direction  of  the  normal  is 
invariable,  and  the  total  amount  of  normal  action  exerted  by  any 
portion  of  the  plane  is  the  same  with  the  ^projection  of  the  tvhole  action 
of  this  portion  of  the  plane  upon  the  perpendicular  to  the  plane,  2vhich  is 
therefore  proportional  to  the  solid  angle  subtended  by  the  portion  of  the 
plane  at  the  point  of  action. 

135.  If  the  surface  returns  into  itself  so  as  to  include  a  space,  vMch 
is  called  a  closed  surface,  and  if  the  point  of  action  is  situated  within  the 
inclosed  space,  the  tvhole  angle  suUended  is  the  entire  extent  of  four  right 
angles  ;  tvhereas,  if  the  point  of  action  is  exteiior  to  the  closed  surface,  the 
tvhole  angle  vanishes ;  hut  it  is  tu'o  right  angles  when  the  point  is  upon  the 
surface.  For,  however  the  point  of  action  is  situated,  if  a  line  is 
drawn  from  it  so  as  to  cut  the  surface  more  than  once,  the 
successive  angles  which  the  line  makes  with  the  exterior  normal, 
will  be  alternately  obtuse  and  acute  as  the  line  cuts  into  the 
surface  or  out  from  it.  The  last  angle,  or  that  of  which  the  vertex 
is  most  remote  from  the  point  of  action  will  always  be  acute.  The 
normal  actions  of  two  successive  elements,  therefore,  upon  the  same 
line,  and  which  subtend  the  same  solid  angle,  are  equal,  but   of 


—  GO  — 

opposite  signs,  so  that  tliey  cancel  each  other's  effect  in  the  total 
sum  of  the  normal  forces.  But  if  the  point  of  action  is  without  the 
surface,  the  first  angle  is  obtuse  upon  each  line,  and  as  the  last 
angle  is  acute,  the  whole  number  of  intersections  is  even,  and  each 
normal  elementary  action  is  cancelled  by  another,  and  the  whole 
sum  vanishes.  If  the  point  of  action  is  within  the  surface,  the  first 
angle  is  acute,  if  there  is  more  than  one ;  and  there  are  an  odd 
number  of  intersections  for  every  direction  in  which  a  line  can 
be  drawn ;  for  each  direction,  therefore,  one,  and  only  one,  normal 
action  remains  uncancelled,  which  is  proportional  to  the  elemen- 
tary solid  angle  ;  and  the  whole  sum  is  that  of  the  entire  extent 
of  four  right  angles.  But,  if  the  point  of  action  is  upon  the 
surfiice,  and  a  tangent  plane  to  the  surface  is  drawn  through  it ; 
every  line  which  is  drawn  from  the  point  upon  the  exterior  side  of 
the  plane  must  cut  the  surface  an  even  number  of  times,  if  it  cuts 
at  all,  precisely  as  if  it  were  drawn  from  an  exterior  point;  but 
every  line  which  is  drawn  upon  the  interior  side  of  the  plane  cuts 
the  surfiice,  as  if  it  w^ere  drawn  from  an  interior  point ;  the  total 
sum,  then,  of  the  uncancelled  elementarj^  solid  angles  includes  those 
for  all  directions  which  are  upon  the  inner  side  of  the  plane,  that  is, 
it  is  equal  to  two  right  angles.  This  elegant  theorem,  given  by 
Gauss,  is  expressed  analytically  in  the  form 


X 


r  4:7t  foi*  a  point  interior  to  a  closed  surface, 
—  =  <  27t  for  a  point  upon  the  surface, 


f 

'     0  for  an  exterior  point. 


136.  The  expression  (SOa)  represents  the  component  in  the 
direction  of  the  external  normal  to  a  surface,  of  the  action  upon  the 
element  of  the  surface  of  a  mass  /c  concentrated  at  the  point  which, 
in  that  expression,  was  the  point  of  action.  The  integral  of  this 
expression  is  the  whole  amount  of  such  resolved   action,  and  by 


—  61  — 
(G02i)  its  value  is 

r —  A: 7th  when  the  mass  /.;  is  interior  to  the  surface, 
—  /       .,^  Jc  =:  —  /    Jc  =  -I  —  2nk  when  the  mass  Jc  is  upon  the  surface, 

^  (^0  when  the  mass  h  is  exterior  to  the  surface. 

Neither  of  these  values  depends  upon  the  position  of  the  acting 
mass  further  than  it  is  interior  or  exterior  to  the  surface  or  upon 
the  surface.     If,  then, 

3Ii  =  all  the  mass  interior  to  the  siurface, 

J[4  =  all  the  mass  upon  the  surface, 

Me  =  all  the  mass  exterior  to  the  surface ; 

tJie  expression  for  the  total  action  of  the  sum  of  all  the  masses  upon  a  closed 
surface,  resolved  for  each  element  in  the  direction  of  the  external  normal,  is 

—  in  31-2.71  M,r, 

and  if  all  the  masses  are  exterior  to  the  surface,  this  sum  vanishes.  If  the 
closed  surface  is  one  of  the  level  surfaces  of  the  system  of  hodies,  this  sum 
expresses  the  total  attraction  of  the  masses  upon  the  surface.  This  impor- 
tant theorem  is  due  to  Gauss,  and,  independently  to  Chasles,  in 
almost  its  full  extent,  as  well  as  most  of  the  following  deductions. 
It  is  applicable,  even  if  the  surface  have  sharp  angles,  because  the 
extent  of  surface  occupied  by  such  angles  is  zero. 

137.  If  the  closed  surface  is  one  of  the  level  surfaces  of  a 
system  of  bodies,  but  not  the  outer  boundary  of  a  space  in  which 
the  potentials  constant,  the  potential  must  at  each  point,  by  §  67, 
increase  in  passing  from  the  interior  to  the  exterior  or  the  reverse, 
so  that  in  this  case  the  sum  (Glie)  does  not  vanish.  But  the  term 
of  this  sum,  which  depends  upon  the  mass  at  the  surface,  may  be 
neo-lected  at  will ;  for  the  whole  mass  of  a  true  geometrical  surface 
is  absolutely  nothing.     Hence,  every  level  surface  must  inclose  masses  of 


—  62  — 

matter,  unless  it  he  the  outer  material  houndary  of  a  space  in  loldch  the 
potential  is  constant. 

138.  When  any  masses  lie  upon  the  closed  surface,  the  geo- 
metrical surface  may,  as  Gauss  observed,  be  arbitrarily  assumed  as 
being  just  exterior  or  interior  to  the  masses,  or  passing  through 
them.  If,  therefore,  all  the  masses  are  so  distributed  upon  a  surface  that  it 
becomes  itself  a  level  surface,  the  potential  is  constant  for  all  the  inclosed 
space,  and  there  is  no  tendency  to  motion  throughout  this  space. 

139.  Around  every  point  of  maximum  or  minimum  potential 
a  level  surface  of  infinitesimal  dimensions  may  obviously  be  drawn ; 
and,  therefore,  every  point  of  maximum  or  minimum  potential  must  be  itself 
a  centre  of  action,  and  cannot  be  a  void  space. 

In  an  inclosed  space,  therefore,  no  point  can  be  found  for  tvhich  the 
value  of  the  'potential  exceeds  the  limits  of  value  tvhich  are  found  upon  the 
inclosing  material  surface;  and  in  no  point  of  unbounded  space  has  the 
potential  so  great  a  value  as  its  greatest  value  ujoon  the  exterior  surface  of 
the  finite  masses.     This  inference  was  drawn  by  Gauss. 

140.  In  a  system  of  bodies,  of  tvhich  gravitation  is  the  only  force, 
there  can  be  no  point  of  absolute  minimum  p)otential.  For  if  about  a  point 
of  maximum  or  minimum  potential,  as  a  centre,  an  infinitesimal 
sphere  is  described,  there  can  be  no  point  within  the  sphere,  either 
of  maximum  or  minimum  potential,  wdth  reference  to  the  matter 
external  to  the  sphere.  But,  with  reference  to  the  matter  of  the 
sphere  itself,  the  centre  must  be  a  point  of  maximum  potential,  and, 
therefore,  cannot  be  a  point  of  minimum  potential,  with  reference 
to  the  combined  action  of  all  the  masses.  ** 

This  theorem  is  equally  applicable  to  an  aggregation  of  elec- 
tricity, all  of  wdiich  is  of  the  same  kind,  that  is,  wdiich  is  homogeneous 
when  the  point  of  action  is  assumed  to  be  of  the  opposite  kind  of 
electricity. 

141.  If  any  extent  of  level  surface  is  assumed  at  will  as  a 


—  63  — 

base,  and  if  trajectories,  like  those  of  §  68,  are  drawn  through  each 
point  of  its  perimeter,  their  nnion  forms  a  canal.  The  same  canal 
cuts  a  base,  like  the  assumed  base,  from  each  level  surface  which  it 
intersects.  Of  any  canal,  tJien,  wJiich  is  not  extended  so  far  as  to  include 
portions  of  the  attracting  masses,  the  attractions  upon  all  the  hases  are  equal. 
For  the  whole  amount  of  action,  resolved  in  the  direction  of  the 
external  normal,  at  each  point  of  action  upon  the  closed  surface, 
formed  by  the  faces  of  the  canal  and  the  two  terminating  bases, 
vanishes,  because  there  is  no  included  mass.  But  there  is  no  action 
perpendicular  to  the  faces,  that  is,  in  the  direction  of  the  level  sur- 
faces ;  whereas  the  whole  action  upon  the  bases  is  normal  to  them. 
The  actions  uj)on  one  base  are  in  the  directions  of  its  external 
normals,  while  those  upon  the  other  base  are  in  the  directions  of 
the  internal  normals ;  but  these  actions  balance  each  other  in  the 
algebraic  sum,  and,  therefore,  their  absolute  values  must  be  the 
same.  This  theorem  belongs  to  Chasles,  but  the  brief  demonstra- 
tion is  original. 

142.  In  the  follow^ing  simple  view  of  this  whole  subject,  many 
of  its  propositions  are  condensed  into  a  small  compass.  Each  centre 
of  action  may  be  regarded  as  a  fountain  from  which  a  stream  is 
perpetually  flowing  in  every  direction,  with  an  amount  of  discharge 
proportioned  to  the  intensity  of  the  action.  The  quantity  which 
flows  from  each  centre,  for  an  instant,  through  any  given  elemen- 
tary surface,  may  easily  be  shown  to  be  in  exact  proportion  to  the 
force  with  which  the  surface  is  attracted  by  this  centre  perpendicu- 
larly to  itself  and  against  the  current ;  and  that  which  is  true  for 
each  centre  is  also  applicable  to  the  combined  action  of  all  the 
centres.  Upon  a  space,  then,  in  which  there  is  no  spring,  the 
amount  which  is  flowing  out  must  constantly  be  equal  to  that  which 
is  flowing  in  -,  while  from  a  space  which  contains  springs,  the  amount 
which  is  discharged  must  exceed  the  inward  flow  by  all  which  is 


—  64  — 

supplied  by  the  fountains.  These  propositions  are  equivalent  to 
those  of  §  136,  and  it  may  be  shown  by  an  easy  argument  that 
Laplace's  equation,  with  its  modification,  is  merely  the  same  propo- 
sition applied  to  the  element  of  space. 

By  the  additional  hypothesis,  that,  to  preserve  the  uniform 
flow  of  the  stream,  its  density  must  decrease  in  each  element  of 
the  stream  with  the  distance  from  the  origin,  so  as  always  to  be 
inversely  proportional  to  the  distance  from  the  centre,  the  potential 
represents  the  density  of  the  combined  streams,  and  the  level 
surfaces  become  surfaces  of  equal  density.  The  aggregate  current 
of  the  combined  streams  is  also  equivalent  to  a  single  current  in  a 
direction  perpendicular  to  the  level  surfaces,  and  having  a  velocity 
proportionate  to  the  rate  of  decrease  of  density.  But  this  is  the 
well  known  law  of  the  propagation  of  heat,  w^lien  there  is  no 
radiation,  and  hence  arise  the  analogies  between  the  level  and 
isothermal  surfaces,  and  the  identity  of  the  mathematical  investi- 
gations of  the  attractions  of  bodies  and  of  the  propagation  of  heat 
which  have  been  developed  by  Chasles. 

143.  If  an  infinitelij  thin  homogeneous  shell  is  formed  upon  each  level 
surface  of  a  system  of  hodies,  having  at  each  point  a  thickness  jyifoportional 
to  the  attraction  at  that  point,  the  portion  of  cither  of  these  shells,  ivhich  is 
included  in  a  canal  formed  hy  trajectories,  hears  the  same  ratio  to  the  tvhole 
shell,  ivhich  the  portion  of  another  shell  included  in  the  same  canal  hears  to 
that  shell,  provided  there  is  no  mass  included  hetween  the  shells.  For  if  the 
bases  of  the  canal  are  infinitely  small,  they  must  be  reciprocally 
proportional  to  the  intensities  of  the  actions  upon  them,  because  the 
whole  amount  of  action  upon  the  different  bases  is  the  same.  But 
the  thicknesses  of  the  shells  are  proportional  to  the  intensities  of 
action,  and,  therefore,  the  products  of  the  bases  multiplied  by  the 
thicknesses,  or  the  volumes  of  the  portions  of  shell  included  in  the 
same  canal,  bear  a  constant  ratio  to  each  other.     Since  the  ratios 


—  65  — 

are  constant  the  infinitesimal  volumes  may  be  added  together,  and 
their  sums,  which  are  the  volumes  included  in  a  finite  canal,  are  in 
the  same  ratio,  and  these  sums  may  even  be  extended  so  as  to 
include  the  whole  of  each  shell.  Hence  the  volume  of  each  portion 
is  the  same  fractional  part  of  the  volume  of  the  shell  to  which  it 
belongs ;  and,  as  each  shell  is  homogeneous,  the  mass  of  each  por- 
tion is  the  same  fractional  part  of  the  mass  of  the  whole  shell.  The 
conception  of  these  shells,  and  the  investigation  of  their  acting  and 
reacting  properties  was  original  with  Chasles,  and  it  will  be  con- 
venient, as  it  is  appropriate,  to  designate  them  as  Chaslesian  shells. 

144.  The  volume  or  mass  of  a  Chaslesian  shell  has  a  simple 
ratio  to  the  attracting  mass  included  within  it,  dependent  upon  its 
own  density  and  thickness.  For  each  infinitesimal  element  of  its 
volume  or  mass  is  proportional  to  the  product  of  the  element  of  the 
surface  multiplied  by  the  thickness  of  the  shell,  and  the  thickness  at 
each  point  is  proportional  to  the  attraction  at  that  point.  The  sum 
of  all  the  elements,  therefore,  of  either  volume  or  mass,  that  is,  the 
wdiole  volume  or  mass,  is  proportional  to  the  sum  of  all  the  attrac- 
tions upon  the  whole  surface.  But,  by  §  136,  the  sum  of  all  the 
attractions  upon  the  surface  is  proportional  to  the  included  mass,  if 
there  is  no  mass  at  the  surface.     If,  then, 

II  is  the  volume  of  the  shell, 
Jc  its  density, 

h  the  modulus  of  its  thickness,  or  the  thickness  which  corre- 
sponds to  the  unit  of  attraction  ; 

this  ratio  is  included  in  the  equation 

-^  —  -^—471- 

145.  If  a  Chaslesian  shell  ivhich  is  ivholly  external   to  the  acting 
masses  of  the  system  is  assumed  to  he  itself  the  attracting  mass  ; 

9 


—  GG  — 

1.  The  potential  of  the  shell  is  constant  for  all  interior  points,  there 
is  no  tendency  to  motion  tvithin  it,  and  its  02vn  older  surface  is  its  level 
surface  ; 

2.  Its  external  level  surfaces  are  the  same  as  those  of  the  original 
masses  of  the  system,  and  the  attraction  of  the  shell  upon  a  point  external  to 
itself  has  the  same  direction  as  the  cdtraction  of  the  original  masses. 

To  demonstrate  these  propositions,  let 

i2^  be  the  potential  of  the  shell  for  any  point,  and 
£2  the  potential  of  the  original  masses  for  each  point  of  the 
shell  ; 

the  value  of  the  element  of  the  potential  of  the  shell  is 


Hence, 


In  passing  along  the  canal  of  the  trajectories  to  another  shell, 
the  ratio  of  dji  to  f/  is,  by  §  143,  constant,  whence 

But 


dS2, 

_  led  11 

~  f 

dS2, 

Idn 

n 

~    t^f 

d^i  ^hdo  Djs;i2  • 


and,  therefore, 


diiD^f=  —  hdo  D^S2  D^Ngo^  f  =:  —  hdo  D^il  cosy , 
-p.  d  i2,, hli  D-^  SI  cos  ^da 

The   integral   of  this   equation  for  the  whole    surface    of  the 


—  71  — 

Newtonian  shell,  and  at  the  corresponding  points  there  will  be 
corresponding  elements  of  volume. 

151.  The  cotT'esponding  elements  of  volume  or  mass  of  two  corre- 
sponding Netvlonian  shells  are  proportional  to  the  volumes  or  masses  of  the 
shells.     For  if 

A-^,  Ay,  A^  are  the  semiaxes  of  the  outer  ellipsoidal  surface  of 

one  shell, 
B^,  By,  B^  those  of  its  inner  ellipsoidal  surface, 
o  its  volume, 
m  its  mass,  and 
n  the  ratio  of  either  axis  of  the  inner  surface,  divided  by  the 

corresponding  axis  of  the  outer  surface  ; 

and  if  the  same  letters  accented  denote  the  same  quantities  for  the 
corresponding  shell,  the  construction  of  the  shells  gives  for  each 
axis 

B^  =  nA^, 

X  x' 


A       A' J 

and 

n  =  ?/ ', 

and  by  differentiation, 

dx         Ax 
dx'        A'' 

The   volumes    and    masses   are    by   well-known   theorems   of 
geometry 

m  ■=:Jco  =^^  71  k  {A^ Ay A^  —  B^ By B^) 

=  i-7ik{l  —  n^)A,AyA,, 
m'  =  I/o'=iTik'{l  —  n')A',A'yA:. 

The  ratios  of  the  elements  of  volume  and  mass  are,  then, 
da  dxdydz    A^A,,A^ a 


da'         dx'dy'dz!         A'^A'^A 


'  A' /t" 


—  TJ^  — 
dm         hda         ha         m 


dm'         kda'         ko'         m'' 

152.  If  the  Older  surfaces  of  two  corresponding' Newtonian  shells  have 
the  same  foci,  their  inner  surfaces  must  also  have  the  same  foci.     For  if 

€  ^  is  the  difference  of  the  squares  of  the  corresponding  axes  of 
the  outer  surfaces, 

the  condition  of  the  identity  of  foci  gives  the  equations 

,'^  =  ^^  —  A':^=:Al  —  A\^=..^^  —  A7. 

HencC;  for  each  axis,  there  is  the  equation 

Bl -  B^  =  n'^{A!  -  A7)  =  n'e\ 

so  that  the  foci  of  tlie  inner  surfaces  are  also  identical. 

153.  If  the  radius  vector,  from  the  centre  of  any  point  of  an  ellipsoid, 
is  projected  upon  the  radius  vector  of  another  ellipsoid  ivhich  has  the  same 
foci,  and  if  the  radius  vector  of  the  corresponding  point  of  the  second  ellipsoid 

is  projected  upon  that  radius  vector  of  the  first  ellipsoid,  tvhich  corresponds  hi 
direction  to  the  projeciion  in  the  second  ellipsoid,  the  two  projections  are 
inverselg  p)roportional  to  the  radii  vectores  upon  tvhich  they  are  p)rojected. 
For  if 

Q  is  the  radius  vector  of  the  first  ellipsoid  ujoon  which   the 

projection  is  made,  and 
'i,  1],  L,  are  the  coordinates  of  the  extremity  of  q  ; 

the  equations  of  the  corresponding  points  give,  for  each  axis, 


whence 


r 

X 

x' 

A' 

A 

A 

5 

1 

r 

—    __ 

-x" 

X 

<o 


or 

But  if 

p  is  the  projection  of/  upon  (>,  and 
j/  the  projection  of  r  upon  (^>', 


these  projections  are 


whence 


J-  Q  (>  (>  Q 


p'  —  Q 


154.  The  difference  of  the  squares  of  ilie  radii  vector es  from  the 
centre^  of  two  corresponding  points  upon  the  surface  of  two  ellipsoids  tvhich 
have  the  same  foci,  is  equal  to  the  difference  of  the  squares  of  their  semiaxes. 
For  the  equations  of  these  surfaces  are 

2                                          /■'' 
NT-^   1  ^_  1 

The  difference  of  the  squares  of  two  corresponding  radii 
vectores  for  j)oints  at  the  surface,  is 

,^ -  r-  =  2^{.^-.-)  =  2, [.r  =  (l  -  f )] 

155.  The  distance  of  any  point  upon  the  surface  of  an  ellipsoid, 
from  a  point  upon  the  surface  of  another  ellipsoid  tvhich  has  the  same  foci, 
is  equal  to  the  distance  of  the  two  corresponding  points  of  the  ellipsoids  from 
each  other.     For  if 

10 


—  74  — 

/  is  the  distance  of  the  point  of  which 
cc,  ?/,  z^  are  the  coordinates,  from  the  point  of  which 
i',  i/,  t' ,  are  the  coordinates,  and 
f  the  distance  of  the  corresponding  points ; 

the  values  of  these  distances  become,  by  (Tog.io)  and  (73.24), 

'2 9//,/ 


p=r'+e-'^u 


*>  21  '2       1  2  i)      I     I 

9     1       /9  o    '   ' y2 

whence 

156.  The  external  level  mrf aces  of  an  ellipsoidal  Chaslesiau  shell  are 
those  of  ellipsoids  ivhich  have  the  same  foci  ivith  the  outer  surface  of  the 
Chaslesiau  shell.     For  if 

SI,  is  the  potential  of  the  given  shell  for  any  point  of  the 
external  ellipsoidal  surface  of  the  same  foci,  and 

11[  the  constant  value  of  the  potential  of  the  corresponding 
Chaslesian  shell,  constructed  upon  the  external  ellipsoidal 
surface,  for  any  internal  point,  and,  therefore,  for  any 
point  of  the  surface  of  the  given  shell ; 

the  equations  (72^)  and  (74i2)  give 
o 


The  value  of  £2,  is,  therefore,  constant  for  all  points  of  the 
surface  of  the  external  ellipsoid,  so  that  this  is  one  of  the  level 
surfaces  of  the  given  shell. 

157.    The  attractions  of  two  corresponding  Neiiionian  shells,  which  have 


—  75  — 

ilie  same  foci,  upon  an  external  point,  have  ihe  same  direction,  and  are  propor- 
iional  to  the  masses  of  the  shells.  For  the  infinitely  thin  shell,  this 
proposition  is  a  simple  corollary  from  (742,5).  But  the  finite  shells 
can  be  subdivided  into  corresponding  infinitesimal  shells,  and  the 
masses  of  the  corresponding  elementary  shells  will  be  proportional 
to  the  masses  of  their  respective  finite  shells.  The  attractions  of 
the  corresponding  elementary  shells  upon  an  external  point,  there- 
fore, coincide  in  direction,  and  are  proportional  to  the  masses  of  the 
shells;  and,  therefore,  the  components  of  all  the  corresponding 
attractions  have  the  same  common  ratio,  and  coincide  in  direction. 
But  the  components  of  all  the  elementary  attractions  constitute  the 
attractions  of  the  finite  shells  themselves.  Several  special  cases  of 
this  theorem  were  first  given  by  Maclaurin,  but  the  general  form 
was  first  demonstrated  by  Laplace,  and  afterwards  more  rigorously 
by  Legendre,  and  it  includes  the  case  in  ivliich  the  inner  surfaces  are  reduced 
to  the  centred  point,  and  the  shells  hecome  ellipsoids,  having  the  same  foci. 

158.  The  attraction  of  any  Chaslesian  shell  upon  a  point  at  its 
surface  is,  from  its  construction,  perpendicular  to  the  surflice,  and 
proportional  to  the  thickness  of  the  shell  at  that  point.  The  attrac- 
tion upon  the  whole  surflice  is,  therefore,  proportional  to  the  mass 
of  the  surface,  which  corresponds  to  §  136.     Hence,  if 

dN  is  the  thickness  at  any  point,  and 

p  the  perpendicular  from  the  centre  upon  the  tangent  plane  at 
that  point, 

the  attraction  of  the  ellipsoidal  Chaslesian  shell  at  the  point  is 

4  Tt  Z"  dN  =  4  TT  Z:  dr  cos  ^. 
■=.  4>'t/:— rcos 

;•  p 

A  7         ^^r 

=  4:71 /Cp—r-, 


—  76  — 
The  component  of  this  action  in  the  direction  of  the  axis  of  x  is 

47rA"»-i— cos     . 
If,  moreover,  the  equation  of  the  ellipsoid  is 

the  general  theory  of  contact  gives 


N         D,L  pD,L 

cos     —  — 


X— ^(QZ)         2:,{xB,L) 


Hence, 


2x 

2 


D.L^'4, 


^-UL—^  - 


2,xD,L  =  2Z,^,=  'l, 


-■A 


N      px 


and  the  attraction  in  the  direction  of  the  axis  of  x  of  the  ellipsoidal  Chasle- 
sian  shell  upon  a  point  at  its  surface  is 

'inkp^x—T^. 

159.  The  attraction  of  an  ellipsoidal  Chaslesian  shell  upon 
any  external  point  is  obtained  by  describing  the  corresponding 
Chaslesian  shell,  for  which  this  point  is  upon  the  outer  surface,  and 
the  attractions  of  the  two  shells  for  this  point  have  the  same  direc- 
tion, and  are  proportional  to  their  masses ;  so  that  the  attractions 
in  any  direction  are  proportional  to  the  masses.  If,  then,  the 
accented  letters  refer  to  the  outer  shell,  the  attraction  of  the  inner 


—  77  — 
shell  is 

IGO.  The  condition  that  the  outer  surface  of  the  exterior  shell 
passes  through  the  attracted  point,  is  expressed  by  the  equation 

This  is  an  equation  of  the  third  degree  when  it  is  reduced  to 
its  simplest  form.  But  there  are  two  other  surfaces  which  can  be 
drawn  through  the  given  point,  and  which  depend  for  their  defini- 
tion upon  the  solution  of  the  same  equation.  They  are  two 
hyperboloids,  both  of  which  have  the  same  foci  with  the  outer 
surface  of  the  inner  shell,  one  of  which  is  a  bipartite,  and  the  other 
an  unparted  hyperboloid.  For  each  of  the  hyperboloids  £^  is 
negative,  and  its  absolute  value,  independent  of  its  sign,  is  contained, 
in  the  case  of  the  unparted  hyperboloid,  between  the  squares  of  the 
mean  and  least  axes  of  the  given  ellipsoid,  and,  in  the  case  of  the 
biparted  hyperboloid,  between  the  squares  of  the  mean  and  greatest 
axes. 

161.  The  points  in  ivhich  all  the  ellipsoids,  ivhich  have  the  same  foci, 
are  cut  hj  the  common  intersection  of  the  two  hjperholoids  ivhich  have  the 
same  foci,  are  corresponding  points.     For  if 

£'^  is  the  value  of  —  £^  for  either  hyperboloid, 

the  equation  of  the  hyperboloid  for  the  points  of  intersection  with 
the  ellipsoid  is 


'A'.  —  i!' 


If  the  equation  (77;)  of  the   ellipsoid  is  subtracted  from  this 
equation,  the  remainder  divided  by  e^  -f-  ^'^  is 


—  78  — 


r.'-^ 


in  which  /,  y\  and  z  are  accented,  in  order  not  to  interfere  with 
the  notation  which  has  been  adopted  for  the  corresponding  points, 
and  which  gives  for  each  axis 

X  x'  x! 


The  substitution  of  these  equations  in  (78i)  reduces  it  to 

7^  =  0; 


the  2:)roduct  of  which  by  fc'^,  added  to  (TGq),  is 


1, 


---AliAi-^^)—      ^Ai  —  ^^ 

which  expresses  that  the  point  [x,y,z)  is  upon  the  surface  of  the 
hyperboloid,  and,  therefore,  all  the  corresponding  points  are  upon 
the  surfaces  of  both  hyperboloids. 

1G2.  The  hf/perholoids  and  elKjJSoids  ivhich  have  the  same  foci,  inter- 
sect each  other  loerpendicularly .  The  conditions  that  two  surfaces  of 
which  the  equations  are 

Z  =  0,  and  ^=0, 

intersect  each  other  perpendicularly  is  expressed  algebraically  by 
the  equation  for  each  point  of  the  line  of  intersection. 

But  for  the  hyperboloids  of  equation  (7728)  ^^d  the  ellipsoid  of 
equation  (76e)  this  condition  becomes 

Ai{Ai.  —  £ -)         ' 
which  is  the  same  with  the  equation  already  given  in  (78io).     This 


—  79  — 

same  demonstration  may  be  applied  to  the  condition  of  the  perpen- 
dicuhirity  of  the  hyperboloids,  if  Al  is  diminished  by  fc'^,  and  t^  is 
changed  into  the  difference  of  the  squares  of  the  semiaxes  of  the 
two  hyperboloids. 

163.  It  follows  from  these  two  theorems,  which  are  derived 
from  Chasles,  that  each  normal  transversal  to  the  ellipsoidal  surfaces  of 
level  is  the  line  of  intersection  of  two  hfjperholoids  tvhich  have  the  same  foci. 

164.  The  lines  of  intersection  of  these  three  surfaces  are,  upon 
each  surface,  the  lines  of  greatest  and  least  curvature,  for  they  are  a 
special  case  of  the  theorem  demonstrated  geometrically  by  Dupin, 
that  the  intersections  of  three  surfaces  ivhich  cut  each  other  at  right  angles 
at  and  infinitely  near  their  common  point  of  intersection,  are  their  lines  of 
greatest  and  least  curvature  at  this  point.  To  demonstrate  this  theorem, 
let  the  three  normals  to  the  three  surfaces  at  the  common  point  of 
intersection  be  assumed  for  the  axes  of  rectangular  coordinates,  and 
let 

be  the  equation  of  the  surface,  which  is  perpendicular  to  the  axis  of 
X.     This  condition  gives  for  either  of  the  other  two  axes 

in  which  equation  r^•,^,  and  z  may  be  mutually  interchanged,  except 
that  the  same  axial  letter  must  not  be  repeated  in  the  equation. 
Those  equations  satisfy  of  themselves  the  condition  (T825)  of  per- 
pendicularity of  these  surfaces  at  the  point  of  intersection.  But  the 
intersection  of  any  two  of  these  surfaces  coincides  w^ith  the  axis 
which  is  the  intersection  of  their  tangent  planes  for  an  infinitesimal 
distance,  and  the  two  surfaces  are  perpendicular  to  each  other  for 
this  distance.  Hence,  each  pair  of  surfaces  gives  an  equation  of  the 
form 


—  80  — 
which  is  reduced  by  (792o)  to 

The  other  surfaces  give  the  corresponding  equations 

The  sum  of  the  products  obtained  by  multiplying  the  first  of 
these  equations  by  D^L^,  the  second  by  — D^L^,  and  the  third  by 


DyLy        is 


1DM.BMBl.L^^^. 


and  the  corresponding  similar  equations  are  obtained  by  advancing 
each  letter  to  the  following  letter  of  the  series,  rr,  y,  z,  and  x.  But 
the  factors  D^L^,  ByLy,  and  D.L^,  are  not  zero,  and,  therefore, 
these  equations  may  be  reduced  to 

which  are  the  well-known  conditions  that  the  directions  of  the  axes 
ofx,?/,  and  s  respectively  coincide  w^itli  those  of  the  lines  of  greatest 
and  least  curvature  of  the  three  surfaces  at  the  origin. 

165.  The  remarkable  relations  of  these  surfaces  might  be  still 
further  extended,  and  if  it  were  worth  wdiile  to  investigate  the 
attractions  of  masses  of  infinite  extent,  it  might  be  shown  that  upon 
each  series  of  orthogonal  transversal  surfaces,  Chaslesian  shells  of 
infinite  extent  might  be  constructed.  The  level  surfaces  of  these 
shells  would  be  the  orthogonal  transversal  surfaces  of  the  same 
series,  while  their  orthogonal  transversal  surfaces  would  be  the  level 
surfaces  of  the  original  Chaslesian  shells  and  the  other  series  of 
orthogonal  transversal  surfaces. 

1G6.  To  investigate  the  attraction  of  an  ellipsoid  upon  an 
external  point,  it  may  be  supposed  to  be  divided  into  an  infinite 


—  81  — 

series  of  elementary  Cliaslesiaii  shells.     Let  then 

A^,  Ay,  yL,  be  the  semiaxes  of  the  ellipsoid, 
((_,,  a,j,  a.,  those  of  the  outer  surface  of  either  of  the  elementary 
Chaslesian  shells,  and  let 

a ,   a„  a, 

-t'i_t  ^"ly  _tl-. 

If,  moreover,  .r,  y,  .t,  are  the  coordinates  of  the  attracted  point, 

A'_^,  Ay,  A[,  are  the  semiaxes  of  the  ellipsoid,  which  has  the 
same  foci  with  the  given  ellipsoid,  and  whose  surface 
passes  through  the  attracted  point, 

u^,  ciy,  a.,  the  semiaxes  of  the  ellipsoidal  surface,  corre- 
sponding to  the  outer  surflice  of  the  Chaslesian  shell, 
and  passing  through  the  point  of  action, 

JE^  =  A'j}  —  A1,,  and 
£  --  ;r  =  a^'  —  «:^  =  (1/  —  A^_^n-; 

the  values  of  E  and  £  are  the  roots  of  the  equations 

S         ^'       =-^        '''       —  1 

The  attraction  of  the  Chaslesian  shell  upon  the  external  point 
in  the  direction  of  the  axis  of  .r  is  by  {11 2) 

^^'^•^  ~zrry~T  •  ~:~  —  -tTi  kx     ,  3  ,   ,  .  —  , 


a^/UyCi',       a  J.  a'^^a'ijCi'.  '  n 


in  which   the  value  of  p    is,   by  equations   (TOg^ig),  given    in    the 
form 

1  _  UL         _  ^  -' 


/^  — [2-(xA.X)]^ 


X-  ^     ^ 

~1     iyl  ~T        a 


11 


—  82  — 
The  differential  of  (SUi)  after  it  is  multiplied  by  ir  is 

^vlience  by  (8I31) 

11  ^ 

(In  = -,'cdt . 

V  - 

This  value  reduces  the  attraction  of  the  shell  in  the  direction 
of  the  axis  of  .f  to 

—  4 71  lex  ' ,3  ;  ,  t at,  =  —  J ;t /c.v 


^[{Ai^ey{AlJ^e^){Ai^,^)y 


The  integral  of  this  expression  is  the  attraction  of  the  whole 
ellipsoid.  The  limits  of  integration  correspond  to  the  values  of  t, 
for  one  of  which  the  shell  is  evanescent,  and  for  the  other  its  surftice 
coincides  wdth  the  surface  of  the  ellipsoid.  But,  when  the  shell 
vanishes,  n  is  zero,  and  e  is  infinite  ;  and  when  its  outer  surface 
coincides  with  that  of  the  ellipsoid,  u  is  unity,  and  t  becomes  E. 
Hence,  the  expression  for  the  attraction  of  the  ellipsoid  in  the 
direction  of  the  axis  of  x  is,  if 

M  is  the  mass  of  the  ellipsoid,  and 
/rits  mean  density, 

\  J . 


2Kj^(Al-\-e')s/[{Al  +  e^iAl-i-s^)(Al-^e-)-] 

By  advancing  each  letter  in  the  series  x,  fj,  £■,  and  x  to  the  follow- 
ing, the  corresponding  expressions  are  obtained  for  the  attractions 
in  the  directions  of  the  other  two  axes. 
1G7.     By  the  substitution  of 

h\  =  Al  +  .\ 


—  83  — 
the  equation  (8224)  becomes 

r>  _  ^^  f   ^ 

1G8.     By  the  substitution  of 

?r,  =  ^-,  and 

6,, 

the  equation  (883)  becomes 

j^  f^ ?jMx  r  kill 


KA,.,)u^   sJ[{Ai-^ul{Al  —  At))iAl-\-uliAl  —  A^)y 


Avhich  formula,    Avith  tran.-<formations    similar    to    the    following,    is 
given  by  Legexdre. 

169.     If -4^  is  as.-;umed  to  be  the  greatest  of  the  semiaxes  of  the 
ellipsoid,  and  A-  the  least,  let 

A-2  A2 

sin  ^  =^  sin  /sin  9: , 

,    r  Al  +  E-' 

CO.       1    —  j^.^^1^ 

yin  0  =  sin  /sin  4* ; 

and   let    the   first   and    second    forms   of  the    elliptic    integrals   be 
expressed  by  the  notation 

6 


S^.g)  =z  /  sec (5, 


0  ^ 


^j  (f  =  I  cos  (5 

Jo 
0  ^ 


—  84  — 
These  eqimtions  give 

(^Al-\-■c'')^m''^>  =  Al  —  Al  +  {Al  —  Al)f^mHl>^{Al  —  Al)QOfiH, 
(Al  +  ,'')£n''ip=Al  —  Al, 

d.-c^  =■  —  2(^1  —  Al)  cosec^(j  cos (fd(f  ; 

Avliicli,  substituted  in  (82.24),  reduce  the  expression  for  the  attraction 
in  the  direction  of  the  axis  of  x,  when  the  eUipsoid  is  homogeneous, 
to  the  form 

-I-^    •^  n  71*-      [*   sin -ff  sec  ^.^  o  Mx  i'  ,         ,  ,s 

J  6  i-^  X  —  -^  -;  (A  ,.  —  A'.)  -'  sm  -  i  t/  0 

or,  if  i^  = 


the  attraction  is 

^  sin  -  i  ^      '  ' 

The  same  substitution  gives   the    attractions   parallel    to    the 
other  axes  in  the  forms 

Dy  il  =  Py  I  sin  ^  9  sec  ^  ^ , 

0     ^ 


A-^^  =  P^  Ttan^c/nseci? 

0  ' 


But  the  differential  of  the  logarithm  of  (80.21)  is 

cot(^Z>.<^  =  cote/), 
and,  therefore, 

D.^  =z  tan () cot 9^', 


—  85  — 

D^  ( tan  f/  cos  (.^ )  =  sec  ^  ^  cos  (3  —  sin  ^  t^  sec  (3 

=  sect3(sec^9cos^t3  —  sin-t?) 
=  sec  (3  sec  ^9  —  sec  t^  sin  ^  t^  (sec  -  (/  -)-  1 ) 
=  sec  (3  sec ^9-  —  sec  t^  sin ^/( tan ^f/  -|-  sin-(/) 
=  sect3  -|-  cos ^ / sec (^ tan -9)  —  sect? sin- (3 
=^  cos^  -|-  cos^ /sec  (3  tan -91, 

D   i  sin  q-  cos  u  sec  t3 )  =.  -^4>0'^^^cos(ir)  __  cos  (y  cot  qr  sec  ^  g  tan  d  —  tan  ^  sin  y 
f^'       ^       '  ■'  '  ^  sin?  sini 

=  cos^^)  sec'^t3  —  sect'sin^c^ 
=  —  cos ^ /sin ^f/;  sec'^^  4"  (1  —  sin-(/  sin^/)  sec'^^ 
—  sec  (3  sin  ^f/) 

9  •   •     9  q  <     I  A  /t  sin'"^\ 

=  —  cos-^sm-ff  sec'^d  -I-  sece  ( 1 1 

'  \  sin-// 

=  —  cos-ism^'msec'^ti  + -^^-.(cos-t3  —  cos^O 

■*  '     sni  -i^  ' 

=  —  COS  -  /sin  ^  (p  sec'^  t^  —  cot"/  sec  t'  -|-  cosec"  /  cos  t3 . 
These  equations  reduce  the  attractions  to  the  forms 

rk    f^          7)      r  rsec^ecosi?  —  sec^  "  •  t^   /  •  wl 

Dyll  =  P^J   [ ^.^,. sec-«i>^(sni9cos9- sect^) J 

0  ° 

=  P^(icosec^2/^^«?J  —  cosec^ /?)=,<?>  —  sec^/sincjf>cos<f>sec6)), 

i>^/2  =  sec^/P^  /*  [i>^(tanff.cost3)  —  cos(3] 

=:  sec^/P^- (tan c/^ cos  0  —  ^.fF»)  • 

170.    The  following  values  are  derived  from  (8323_24)  and  (8I15); 

cos2<f>=      ^      — 


sin  2  0 


.1;  — J; A'J  —  A^' 

Ai  +  E-'~        aT~' 
Al  —  Al        A'^  —  A:/ 


Ai-^E-'        A';:" 


SG 


cos--  0  =     ,,     I      j^..,  =  -TT,, 


Sin  ■'  I 


COS"  I 


Al  —  Al^Al  —  A^ 
A  X  —  A  J  A^  —  A^~ 
Al  —  Al        A?  — A';' 


Al  —  Al  —  A:/  —  Ar 


The  equations  of  the  attractions  give^  by  means  of  these  values, 
that  of  Panel  (8I15), 

j^^^^  =:,  2^^D,^I2  =  sec2/PsinfZ'secf/'sec0(cos2r9  —  cos-0) 
SM  ,     M 


This  simple  equation  is  due  to  Legendre,  and  the  first  of  the 
two  following  equations  which  are  obtained  by  the  same  process  of 
reduction. 

171.     By  putting 

X  =  .  f ^ ^ - \ g;  0 

tlie  attractions  may  assume  the  form 

DJl^  —  ^MzB^^  L. 

in  which  the  differentiations,  relatively  to  A\,  A^,  and  Al,  are 
performed  without  regard  to  the  changes  oi  E,  dependent  upon  the 
formula  (Sl^j). 


—  87  — 

172.  The  e(|uatioii  (Sl^.j  may,  by  meaiLs  ui"  the  e(|Liatioiis 
(8^2-7)  'je  written  in  the  form 

x'--^1/Hqg''  0  +  5-.sec--'/'  =  (Al  —  Al)co^ec'-'P, 

or  by  the  substitution  of  the  value  of  0  from  (8824), 

a:-  +  ^-sec2(Z*    ,  y^  1 

Ai  —  Al         '     A:.cos^-0-\-  A  fsin  -  (p  —  A  l         ^m-(p' 

173.  Whcu  the  attracted  ^^oiut  is  iqmn  the  surface  of  the  ellipsoid,  E 
vanishes,  and  the  value  of  <f>  becomes 

cos'f'  =  -^. 

Ax 

174.  VChen  the  attracted  point  is  tvithiii  the  ellipsoid,  the  Newtonian 
shell,  of  which  the  outer  surface  is  that  of  the  ellipsoid,  and  the 
inner  surface  passes  through  the  point,  exerts  no  action  upon  the 
point,  and  the  attraction  is  reduced  to  that  of  an  ellipsoid  similar  to  the 
given  ellipsoid,  and  of  ivhich  the  surf  ace  passes  through  the  attracted  point. 

175.  When  the  density  of  the  ellipsoid  varies  in  its  interior, 
in  such  a  way  that  each  of  its  component  Chaslesian  shells  is  homo- 
geneous, Jc  is  a  function  of  f,  and  after  its  substitution  (8294)  may  be 
integrated. 

176.  When  the  ellipsoid  is  a  homogeneous  ohlate  ellipsoid  of  revolution 
the  various  formulae  become 

^'^+/  +  .^-'  +  ^-'tan^*  =  (.4|-^!)(l+cot^</^), 
zH^w^^  +  (r-  —  .I'i  +  Al)  tan24>  =:  Al  —  Alx 


0  ^ 
I),J2  ^Fgf  sin^c^  =  iP^(2^/'  —  sin2</>). 


88  — 


0  ■ 

177.      When  the  ellipsoid  is  an  homogeneous  prolcde  ellipsoid  of^revolu- 
iion,  the  formulas  become 


A 

= 

Al. 

i 

= 

l^f, 

T 

= 

^, 

D^ll  1=  Px  /  sin^g)secy 

0    ^ 

:=  P;i'  [log tan  {{n  -\-  i  </>)  —  sin «/»] , 

<i> 
D,j£l  =  P u  I  sin^f/'sec^f/) 

0    ' 

==  i  ^U  [sin  f/'  sec^  f/>  —  log  tan  ( ]  ^t:  +  ^  ^Z^)] , 
P-wi  =  P.5'  f  sin^cj}sec^9 

=  I  Pz  [sin  *fJ  sec^  ^  —  log  tan  {\n  -\-  \  f/>)]  . 


ATTRACTION    OF   A    SPHEROID.       LEGENDRe's    AND    LAPLACE's    FUNCTIONS. 

178.  The  investigation  of  the  attraction  of  a  spheroid  is 
greatly  facilitated  by  the  introduction  of  certain  functions  which 
were  first  conceived  and  investigated  by  Legend  re,  but  which 
became  so  fruitfid  in  their  more  general  form,  given  in  the  subse- 
quent researches  of  Laplace,  that  they  are  usually  designated  by 
the  name  of  the  latter  geometer.  A  method  will  be  pursued  in 
their  development  and  discussion  which  is  similar  in  some  respects 
to  that  given  by  Jacobi. 


—  80  — 
170.     Lot 

IIzzzi  COS  w  -\~  /sin fj"  Qo^i] , 

and  if  any  power  of  II,  denoted  hy  n,  is  developed  in  a  series  of 
terms  arranged  according  to  the  cosines  of  the  multiples  of  iy,  let 
any  one  of  the  terms  be  denoted  by 

in  which  [;;^-]  denotes  the  number  of  accents  of  ^7^.  The  required 
power  has  then  the  form 

CO 

IP  =  ^,„  {("'  ^^ir^  cos  m  r^ ) . 

00 

180.  The  value  of  His  not  changed  by  reversing  the  sign  of 
rj,  and,  therefore,  the  series  remains  unchanged  by  this  reversal  of 
sign,  which  gives 

or  ^^-'"^  =  ±  4^1"^  =  (—  l)'"t/>lr^; 

in  which  the  upper  sign  corresponds  to  the  even  values  of  m,  and 
the  lower  sign  to  the  odd  values  of  m.  The  equation  (80n)  nif^y 
also  be  Avritten 

1 

181.  The  integral  of  the  product  of  (8O22)  by  cosniij  is,  by  a 
well  known  theorem 

f  {II"co8mri)  =  27i /'« cPIr^ . 

The  derivative  of  this  equation,  relatively  to  (p,  reduced  by  the 
condition 

D.R=^  —  sin  if)  -f- 1  cos (p  costj , 
12 


—  00  — 

becomes 
9 


n  i"'D^  '^P\r'^  =^n  I  ^B"  ~  ^  cos  ni  i]  ( —  sin  (p  -j-  z  cos  ip  cos  ?y )] 

0 

^=n  I  [ff"~\ — sinf/)cosm>;-|-^?*cos(/(cos(m-|-l)iy-]-cos(?« — l)'i))]  ; 
whence,  by  (SOgr), 

182.  The  derivative  of  (SOgo),  relatively  to  i],  reduced  by  the 
condition 

D^  Hzzzz  —  ^  sin  (p  sin  i] 

becomes 

CO 

2 «  //"  - 1  sin  f/)  sin  ij  =  2  :r„,  (m  /'"  c/>lrJ  sin  ?;^^  >; ) . 
The  integral  of  the  product  of  this  equation  by  sin;«j^  is 

in  sin  (/>  /   ( 7/"'"  -  ^  sin  i;  sin  ?/?  i^ )  =  2  tt  ?« z '"  </>[;"i , 

•  /  7/ 

0 

or 

in  sin  rp  /    [//'"  -  ^  ( l  cos  (w^  —  1 )  i;  —  i-  cos  (?«  4-  1 )  ^y )]  =  2  tt  ?;^  i'"  </>[;"] ; 
0  '' 

which  becomes  by  (SOg;) 

?;« ^^[f']  =  1  ?^  sin  (p  ( ^/'L"!-/]  +  *^lri+i^J) . 

183.  The  equation  (892;)  may  assume  the  form 

j   lIP~\Gos^'Comri-{-lism(pj{cos{m-\-l)ij-^(cos(m — l))']=27ii'" (p^J^\ 

which,  reduced  by  (8%-),  gives 

<P\:'^  =  Qos(p>^P\l"l,  4-  -1  sinf/.(c/>i;'r-/]  —  f/^[r_+/^). 


—  91  — 

184.     The  remainder,  if  (OOg)  is  subtracted  from  the  product 
of  (9O24),  multiplied  by  cotf/,  is 

n  cos (f  4^\;"Jl^^  4- n sin o)  <f^[;"^  1  ==  ?;i  cot  a  <Z^L"'^ — ^n ^^'^f''  =  —  sin"'  (tD  -^ 

=  sin '"  +  19^0.0^- 
The  sum  of  (9O24)  ^'^^^  '^  times  (9O31)  is 

the   first   member    of  Avhich   becomes   identical  with  that   of  the 
previous  equation,  when  m  is  increased  by  unity.     Hence, 

0[:"^^^ 1  ^     0f^    sin  cf.         j^  0^'^ 

sin "'  cff  n  -|-  ;«  -|-  1      9  sin '"  g?         n -\-  m  -\-  1      '^"^  9  sin '"  cp ' 

or,  li  m  is  diminished  by  unity 


sin '"  Cf         n  -\-  in      ^^^  9  sin  '"■    ^  g)  * 

If  the  sign  of  m  is  reversed  in  this  equation,  it  becomes  by 
(89n) 

sin'"9)f?>[;"i  =  ^^Z>,o,^(sin'"+i9)^^[r+^^)- 

185.     It  will  be  found  convenient  here  and  elsewhere  to  adopt 

the  functional  notation 

1 

rh=C(-\og.:c)\ 

0 

which  gives,  by  a  familiar  formula,  or  by  simple   integration  by 
parts,  when  h  is  positive,  and  Jc  an  integer,  which  is  less  than  //  -)-  1, 

1 

rh  =  h (h  —  1) (/^  —  2) (h  —  /c  +  1)  f{—  log.r)^-^- 

0 

=  h{h  —  i){h  —  2) (h  —  k-}~  i)r{h  —  Jc), 


—  02  — 

and 

/,(/,_  l)(/,_2)....(A_Z-+l)  =  ^,^. 

When  h  is  an  integer,  and  h  the  next   smaller   integer,  this 
formnla  becomes 

1.2.3 Ji=rh. 

With  this  notation,  Taylor's  theorem  assumes  the  form 

186.     The  equations  (Oli^)  and  (Olso)  give,  by  successive  sub- 
stitutions in  each  other,  and  the  use  of  the  preceding  notation, 

(—  l)'"r(«  —  m) sin'" 9^  0f;"]  =  (—  iy''r{n  —  m') B':^-''' (sin"'> 4>l:"^) ; 

in  which  negative  differentiation  must  be  interpreted  to  be  integra- 
tion ;  in  the  former  equation,  when  n  is  negative,  m  -\-n-\-l  and 
m  -\-  11  -\-  1  must  be  positive  ;  while,  in  the  latter  equation,  n, 
n  —  m  -f-  1,  and  n  —  m  -\-  1  must  all  be  positive.  When  n  is  posi- 
tive, but  n  —  m  -\-  1,  and  n  —  m  -\-  1  are  negative,  the  equation  to 
be  substituted  for  (92i5)  is 

I\m  —  n  —  l)  r{m'  —  n  —  \)      ' 

which  equation  is  also  to  be  used  when  n  is  negative.  When  n  and 
n  —  m  -\-  1  are  positive,  but  n  —  m  -\-  1  is  negative,  the  combina- 
tion of  (92i5)  and  (9293)  gives,  by  representing  by  n\  the  greatest 
integer  contained  in  n  -f-  1, 

(_  1)  "'-"' r('M  —  m)  sin '"  (f  (//;'0 /■(,/  _  n  —  1)  i);:;;^'"  (sin"'V/,ri;f;"'J) 

/•(«  —  n')  r{m'  —  n—\)  ' 

When  «,  n  -\-  dI  -\-  1,  and    n  -J-  r^i  -\-  1   are    all    negative,    the 


—  9: 


equation  to  be  substituted  for  (92i3)  is 


I\—\~n  —  v\)^\\V"^  l\—\—n  —  iv!)       •=°«'?      siii"''g' 

When  n  and  ?i  -[~  ^'^'+  1  ^^'^  negative,  but  m  -j-  >^  +  1  is  posi- 
tive, the  combination  of  (92i3)  and  (903)  gives,  by  representing  by 
n,  the  greatest  integer  contained  in  —  1  —  ?«, 


D\ 


There  are  pecuUar  considerations  which  simplify  the  investiga- 
tions, when  n  is  integral,  whether  it  be  positive  or  negative  ;  and 
these  are  the  cases  to  which  most  of  the  subsequent  investigations 
are  limited. 

187.  By  reducing  711  or  vi  to  zero,  the  equations  of  the  pre- 
ceding section  give,  for  positive  values  of  «, 

^(—AY'  rnr{m-n-\)  p     ^^ 

V  )      r{n  — 7/)  r(n'  — 71  — l)sm"' cf  J  cos ^       "' 

and  for  negative  values  of  n 

^"     ~  r(— 1— «)sm™(yJcos^      " 

—  r{—l—n)  ^^^^      ^|^cos6^n 

188.  When  n  is  zero,  it  is  easily  seen  that 

11^=1  =  <P„ 
and  that,  for  all  other  values  of  m 


—  94  — 

189.  When  n  is  a  positive  integer,  and  h  is  also  a  positive  integer, 
the  equation  (9l8)  gives 

{2n-\-h)  ^Pl:  +  ^'^  =  n  cos  qi  ^/^L"- /'^  +  n  sin  (f  4^^;,'Zi  +  '^ . 

If,  then,  the  terms  of  the  second  member  vanish  for  any  value 
of??,  they  will  also  vanish  for  the  next  higher  value  of  n.  But  they 
vanish,  by  the  preceding  section,  when  n  is  zero,  and,  therefore,  they 
vanish  for  every  positive  integral  value  of  n  ;  that  is, 

(!)[». +  />]  0 

or  the  series  is  finite  for  ])ositlve  integral  values  of  n,  and  contains  onlg 
n  -\-\  terms. 

190.  The  substitution  of  the  preceding  equation  in  (9l8)  gives 

c/^w  —  x^in(f(p\l'Zx^  =  (1  sinf/))'"</>^fr„T^ 
=  (ising;)'^0o  =  ^sm"9. 

which  equation,  substituted  in  (93i,3),  gives 


fp  = 


(-1Y 


=^"i>;os^(^[r^sin»  =  ^^^L,(-  sin^/O" ; 


=z vzz}zl j)n-m  ( ginS^.yt . 

2T(7i  — m)  sin'" cp     '"'"!'   ^  '^    ' 

bg  loliich  the  coefficients  of  the  development  are  oUained  ivhen  n  is  a  positive 
integer. 

191.      When  n  is  the  negative  of  iinitg,  the  equation  (89.27)  givQs 

0 

But  the  value  of //gives 


-  05  — 

cos  rp  —  ^  sin  rp  cos  ?j 


//         cos  (jp  -|-  ^  sin  q>  cos  //         cos  '^  cp  -j-  sin  -  cp  cos  -  // 

/sinrjpcos?/  ■  cosqp 

1  — sin^^sin^/y    '     cos ^ /; -[- cos "^ g) sin ^ j; 

i-rsinfjfcos//  i^/sin  g^cos?/      .        cos  g  Z),^  tan  »;• 

1  -|-  sin  (f  sin  //         1  —  sin  cp  sin  ij~^  1  -\-  cos  ^  qt^  tan  ^  // ' 

the  inteo-ral  of  which  is 

C  I  T  -1        1 -j- sin  (T- sin  ?7    i     ,       r    n/  j.  \ 

/   77  =  —  l-^  iO^  r^ — .       ■   -  4-  tan  ^~ ^J  ( cos q>  tan  7; ) 

J  yrl  -•        °1  —  smqfsin//     '  ^         '  '^ 

Hence,  by  passing  to  the  limits 

^-1  =  1, 

When  n  and  m  vanish,  equation  (90gi)  becomes 
0^  =:  cos  9J  </^_  1  —  sin  (p  ^P'_  1 , 


whence 


<i*Li  ^ -. =  —  tan  },  (p 

sni  rjf)  "  ' 


Equation  (92ig)  gives,  then, 

- 1  ~         2  I\m  —  1)  ^-^  ^   ^^^    '2  9^  • 

192.  When  n  is  aw/  negative  integer,  it  is  more  convenient  to 
write  the  formulas  with  the  sign  of  n  reversed.  With  this  change, 
the  sum  of  the  product  of  (90gi)  multiplied  by  ( — nm\ip),  that 
of  (908)  multiplied  by  cos  9^,  and  that  of  (9O24)  multiplied  by 
( — cosecg^)  becomes 

.«ci.N+|]^^  =  cos(/)i>^*[^'i  —  (;^sinc/.  +  £^)'^'^'I 

—     secg,    ^^sin"->         V^-"^^         IjSinf/;-t-         ^.^^^       )i-n, 

which,  when 

m  =  «  —  1 , 


—  96  — 
is  reduced  to 


CIJ[ 


11-11 


The  successive  substitution  of  1,  2,  d,  &c.,  for  n,  gives,  by  means 


of  (91,2) 


r  sin  "      qj 


r(2}i)  ,       .     N„ 
The  substitution  of  this  value  in  (9821)  gives,  by  (942o), 

and,  therefore,  for  all  values  of  m  less  than  n  -\-  1, 

The  equation  (9l8)  gives,  when  m  —  n  vanishes,  by  (969), 

whence,  by  (9O24) 

From  this  equation  the  successive  values  of  *?'[!.\j  may  be  deter- 
mined by  successive  substitution  of  1,  2,  3,  &c.,  for  n,  or  they  may 
be  determined  by  the  equation  derived  from  (9I20)  a,nd  (969), 

cos  (i 


—  97  — 

The  remaining  coefficients,  in  wliicli  m  is  greater  than  n,  are 
then  to  be  determined  by  the  equation  derived  from  (92i3) ; 

- "         r{7n  —  n)      """^  <>  sin  >  • 

193.  In  order  to  apply  the  preceding  investigations  to  the 
problem  of  attraction,  it  is  requisite  to  introduce  the  form  of  polar 
coordinates,  of  which  zenith  distance  and  azimuth  is  the  familiar 
instance.     For  this  purpose  let  the  following  notation  be  adojDted : 

^yis  the  angle  which  a  line/ makes  with  the  axis, 
^f  is  the  angle  which  a  plane,  drawn  through  the  axis,  parallel 
to/,  makes  with  the  primitive  plane. 

The  distance  /,  between  two  points,  of  which  the  radii  vectores 
are  r  and  i),  is  given  by  the  equation 

/2  =r  r 2  +  (^) 2  _  2 r ()  (cos c/.,  cos ^^  +  sin ^, sin c/)^ cos ((1,  —  ^p) ) 

=  ^''+^>'  — 2()rcos^. 

Hence  the  notation 

i^sl  —  1, 
Hf  =  cosgiy  -f-  2  sin  c/ycos  {ij  —  ^y) , 
gives 

■^fllf  =  +/COS  f/ly  +  Z/Sin  9)yC0S  (i]  —  ^y) 

^  /'cosf/'^  —  (^cosf/)  -|-  /cosiy(rsin9^cos^^  —  f)sin^  cos^  ) 
-|-  /sin  ri  (r  sin c^, sin  (3^  —  ^  sin  9)  sin ^  ) 

in  which  the  upper  sign  is  to  be  used  when  r  is  greater  than  ^,  and 
the  lower  sign  when  r  is  less  than  (>. 
But  it  folio WS;  from  §  191,  that 


lit  "i- 


0     '  0  ' 

ill  which  the  uj^jper  sign  corresponds  to  r,  greater  than  q,  and  the 
lower  sign  to  r,  less  than  (),  and  the  series  represented  by  the  fourth 
member  corresponds  to  the  former  of  these  two  cases,  while  the 
series  represented  by  the  last  member  corresponds  to  the  latter 
case. 

194.     If,  in  the  development  of  the  preceding  series, 


Q„  is  the  coefficient  of  -^,  and 
Q[,  is  that  of  -^, ; 
the  series  become 

The  values  of  the  coefficient  in  these  series  are  determined  by 
the  equations 

27r 

H'h 


0 
0 

If  the  additional  notation  is  adopted,  corresponding  to  (SOy) 
Hr  =  [/•].  +  2 1.  {i-  [r]L'"^ cos;;^  (;;  —  d,)) , 


1 
the  values  of  the  coefficients  become 


1 

CO 

«.  =  ['■]..  [',']-<«  +  ..  +  2 ^„,((-  1)'"  [r]L"" [(Oi^','..+.co,s™ {i\,  -  ^)). 


—  9'J  — 

Hence,  by  (96,s)  and  (93ii;), 

«.= q:=  Da,[o],.+2|„,(^<"-^;>y;'+"'>  [r];r>  [o]ir'  cos,«  (.^^  -  cv)) 

=['a.[co..+2i4^;^;«''i>s,,;r],.i>r.,,^[-o,.cos„<^-t^^ 

195.     The  equation  (45^)  gives  for  the  value  of  the  potential, 

Hence,  by  the  notation 
the  potential  becomes 

oo  rr  00 

S2  —  :z-^z=::^(  v -/■"). 

0       ^  0 

With  the  notation  of  (0823.27)  ^^^^  (Q'J'io)  these  values  become 
do  ^()^du' d o  =  o 2 sin cppd(fpd^pdQ, 

The  first  form  of  12  in  (99^)  is  to  be  used  for  all  values  of  (^) 
less  than  r,  and  the  second  form  for  all  values  of  o  greater  than  r. 

If,  then,  k  is  supposed  to  vanish  for  all  points  of  space  in  which 
there  is  no  attracting  mass,  the  limits  of  integration  for  the  value  of 
U^  must  include  all  the  attracting  mass  for  which  q  is  less  than  ;•, 
while  those  for  U',  must  include  all  the  attracting  mass  for  which  o 
is  f^reater  than  r. 


—  100  — 

196.  By  substituting  in  (992i_23)  the  values  of  Q,^  given  in 
(992)  the  resulting  values  of  U^  and  U^  have  the  same  form  with 
Qn  so  far  as  the  elements  of  the  direction  of  r  are  involved ;  so  that 
the  value  of  the  term  of  U,^  which  depends  upon  the  angle  m^^  ^^^ 
the  form 

\f]l''^  (  JiL'"^  cos  W^  ^r  +  -^ir^  sill  m  &r)  , 

in  which  A\'"''  and  i?[r^  are  independent  of  the  form  of  the  body, 
and  the  number  of  such  constants  included  in  the  most  general 
value  of  £/"„  is  2  ?2  -]-  1. 

197.  It  is  expedient  to  introduce,  at  this  point  of  the  discus- 
sion, some  important  properties  of  Legendre's  functions.  The 
following  theorem,  given  by  PoissoN,  is  of  especial  use  in  ficilitating 
their  investigation. 

If  Np  denotes  any  function  of  the  elements  of  direction  of  (),  and  if, 
after  the  performance  of  the  integration  expressed  in  the  second  member  of 
the  folloiving  equation,  q  is  made  equal  to  r,  tvhich  condition  is  intended  to  he 
denoted  hj  the  subsequent  parenthesis,  the  second  member  zvill  be  reduced  to 
the  first  member,  that  is, 

4  T 


0  ^ 


To  demonstrate  this  theorem,  it  is  to  be  observed  that  all  the 
elements  of  the  integral  vanish,  except  those  for  which 


that  is,  for  which 


/=o, 


",    =0, 


If,  then, 


1]  denotes  the  angle  which  the  plane  of  r{>  makes  with  any 
assumed  fixed  plane  passing  through  r. 


—  101  — 
the  integral  becomes,  by  (9  Tie), 

1     r  r  ()•■'  — n^-)  Xp iVv    r     r  r  (r^  —  ry-)  i^m] 


27r 


0        0 


198.     The  equation  (9Tio)  gives,  by  means  of  the  first  form  of 
(98,.,), 

r'-'f  =  -p  +  rfDJ  =  -/'  (j,  +  r  a)) 


0 

which  substituted  in  (IOO20)  reduces  it  to 

4- 


*=f-(^-/,"'-''-')=f-^"i-'' 


0 

by  adopting  the  notation 

47r 


0 


in  which  it  must  be  observed  that  when  n  is  zero  it  must  be 
retained  in  the  written  expression  to  avoid  confusion.  It  ma}-  also 
be  remarked  that,  from  the  comparison  of  the  forms  of  (99i2)  and 
(IOI21),  the  most  general  form  of  JS'l"^  is  the  same  uith  that  given  in  §  19G 
for  Z7„. 

199.     If  the  given  function  is  such  that,  for  every  value  of?/ 
different  from  n 

iVTJ  =  o, 


—  102 

the  equations  (101i7_2i)  give 


J 


4.. 


(  0  TV^t"]"!  —  A^f"] 


0    ^ 

tJ  ll) 
0 

The  theorems  expressed  in  the  last  two  equations  are  of  fundamental 
importance,  and  tvere  given  hij  Laplace. 

200.  The  theorem  (IO27),  not  being  limited  to  any  special 
direction  of  r  is  true  for  all  directions ;  and,  therefore,  the  most 
general  form  may  be  substituted  for  Q,,,  which  can  be  obtained  by 
combining  all  its  special  values  in  any  linear  function.  Any  such 
general  form  would  be  the  same  with  that  of  Nf^-,  and  if  it  is 
denoted  for  distinction  by  M\f^,  the  theorem  (102^)  assumes  the  more 
general  form  given  hy  Laplace, 

47r 
0    ^ 

201.  In  considering  the  attraction  of  a  spheroid  upon  an  external 
point,  luhich  is  so  remote  that  r  is  greater  than  ang  value  of  (^  let 

u  be  the  value  of  {)  for  the  surfiice  of  the  spheroid,  and 

CO  1^ 

0  J  P 

0 

the  function  which  is  denoted  by  the  second  member  of  this  equa- 
tion being  developed  in  the  form  of  a  series  of  terms  of  Legendre's 
functions,  by  means  of  (IOI21).     The  equation  (992i)  becomes,  by 


means  of  (IO25), 
and  the  potential  is 


103 


202.  If  the  point  is  so  remote  that  the  squares  of  the  Hnear 
dimensions  of  the  body  may  be  neglected  in  comparison  with  the 
square  of  the  distance  of  the  attracted  point,  it  has  been  shown  in 
§  128  that  the  attraction  is  the  same  as  if  the  body  were  condensed 
upon  its  centre  of  gravity.  In  this  case,  therefore,  if  the  origin  is 
assumed  to  be  the  centre  of  gravity,  the  potential  becomes,  as  in 
(56^), 

III  all  cases,  then,  in  ivhicli  tJie  origin  is  the  centre  of  gravity,  this  erjiia- 
tion  gives  for  an  external  point  tvhich  is  so  remote  that  r  is  greater  than  {>, 

Uq  =  m, 

n  — !?!  _i    ^  (        ^^         VM^ 

203.  A  homogeneous  ellipsoid  can  always  le  found,  of  ivhich  the 
potenticd,  for  any  external  point,  developed  in  the  form  (lOogo),  t^ill  l>e  iden- 
tical tvith  this  expression  in  its  two  first  terms.  To  demonstrate  this 
proposition,  and  develop  the  mode  of  investigating  the  ellipsoid  in  a 
given  case,  it  may  be  observed  that,  if  the  centre  of  the  ellipsoid 
coincides  with  the  centre  of  gravity  of  the  given  spheroid,  and  if 
the  mass  of  the  ellipsoid  is  the  same  with  that  of  the  spheroid,  the 
potential  of  the  ellipsoid,  for  an  external  point,  has  the  form  (lOSso) 
with  the  same  first  term.  The  difficulty  of  the  demonstration  and 
investigation  is  thus  reduced  to  the  consideration   of  the  second 


—  104  — 
term.     The  general  form  of  this  term  is,  by  (OO^)  and  (100c), 

i?[2]  =  A  (cos'c/),  —  J )  +  smf/),cos^,(^cos^,  +  ^'sin^,) 

+  Bmhp,{Ccos2^+  6''sm2^,) 

=  —  ^A-\-  Acos^l  -f-  Clcos^l  —  ^^^^y) 

-[-  -^COS^COS^  -f~  ^  COS^COS^  -J-  J  6'  COS^^COSy 

=  :^, ( (7, (cos^^  —  I)  +  ^.cos^cos'^)  ; 

in  which  last  form  the  arbitrary  constants  C^,  Oy,  C^,  B^,  By,  and 
B.  are  introduced,  for  the  sake  of  symmetry,  and  in  which  the  six 
constants  are  only  equal  to  five,  by  reason  of  the  equation 

JS'_gCOS^^=  1. 

In  the  especial  case  of  Q2,  these  constants  become,  for  the  axis 
of  :r, 

^zrz  Scos^cos?, 

6^  =  2  cos  X  ? 

and  similarly  for  the  axes  of  y  and  ^. 

The  equation  (TGg)  of  the  homogeneous  ellipsoid,  of  which  the 
axes  are  the  given  rectangular  axes,  gives,  for  the  surface  of  this 
ellipsoid, 

1    ,^  cos^^ 

If,  therefore,  K  is  the  density  of  the  ellipsoid,  the  equation 
( 10227)  becomes,  in  this  case, 

u 

00  ^  1 


—  105  — 
and  hence,  by  (1002), 

47r 

*j  1/; 
0  ' 

To  obtain  the  value  of  TJ^^  it  must  be  observed  that,  by  (104i-), 

47r 


because,  from  the  symmetrical  form  of  the  ellipsoid,  the  value  of  ?( 
is  not  altered  by  changing  either  of  the  angles  upon  which  it 
depends  into  its  supplement,  while  the  sign  of  B^^  is  reversed. 

The  remaining  terms  of  the  integral  contained  in  (1062)  have 
the  form 

47r  47r  4  tt 

r  (it'O:)  =  f  r  (it'  cos^  i\  =  I J  (?r^cos2f/v.). 

0    ^  0    ^  0 

But,  by  the  equations 

cos^  =  sine/)  cos ^^,, 


( 10425)  becomes 


cos""  =sincf)  sini^  , 


1   Qos'^cfp    I    s'ln^q^pCos^dp    .    sin  ^  (jryj  sin  - /9p 

u  ^  A  'i     "T"  A  l  "^  A I 

cos- (9,,         sin"^  On 


'% 


Ai    "I"     Al    "I    \^^;  Ai  Al   }     '    > 

by  putting 

rr=K«'  +  yC0s2^p); 

14 


—  lOG  — 

1     ,     1 

a  =1:  -—  -^  --  , 
A'^    '    Ay 

1 
A- 


T  ^  1 


Z*  =  -7^,  —  «. 


The  integral  (lOoij)  is,  therefore, 

47r  2;r         tt 


0    ^  0  0*^ 


•i   TT  TT 


__  3    r       r       cos^psinyp 
^  J  Gp  J  cf^  {a  -\-bcos'^ cpp) 2  ' 

But 


J*       eo-i"^  cf  p  sin  cfp   ^  — cos'qrp 
cpp  {a  -\-  b  cos '  (/  p)  -         3rt  {a  -\-  ^  cos"-  f/p)  - ' 

TT 

r       cos  ^  g-p  sin  g-p  ^  2  2  Az 

J  Cf^  (a-\-b(jos''Cfp)'         oa{a-\-b)-  3  a 

0      ' 


Jd.a       j-20p2a       J2dct'^b\ 


P 
2 


cos  26p 


^       r    1  (a'  —  b')  tan  ^ 


^^,tant-i(^tan<5j, 


i  TT 

J  00^ 


A. A, 


These  values,  with  that  of  the   mass  of  the   ellij)soid,   reduce 
(lOGg)  and  (105o)  to 

4^7r 
"J  lb 

U,  = -1 71  KA^A,A,:^^  ^AUo8''^-iAt) 


lb 
0 


—  107  — 


=  ^\m^^(AUo^'l-iA'^ 


If  the  axes  of  the  ellipsoid  are  not  those  of  :r,^,  2',  but  of /,?/',/, 
this  expression,  by  means  of  the  equation, 


COS_^'  =  COS^COS^^/  -f-  COS    COS ^/-|-  cos^cos"/, 


becomes 


in  which 


^^2  =  /o  "^^4^^' V^^^^  ~~  0  +  ^^'cos^cos^J; 


This  value  becomes  identical,  therefore,  with  that  of  (1048),  if 

/^/^ -^  fi 

■^  dm 

204.  If  the  potential  and  its  component  functions  for  the 
ellipsoid  are  denoted  by  the  letter  e  Avritten  beneath  them,  the 
potential  of  the  spheroid  for  an  external  point,  for  tvhich  r  is  greater  than  o, 
becomes 

n  =  n  +  f,,  [( u^  -  ^„)r-<"+^G . 

e  3  e 

205.  A  transformation  of  coordinates,  which  is  the  reverse  of 
that  by  which  the  equations  (lOTn)  were  obtained  from  the  reference 
of  the  ellipsoid  to  the  axes  of  the  spheroid,  would  bring  the  equa- 
tion (104i9)  to  the  form  (lOTie).  From  the  forms  of  the  expression  it 
is  obvious  that  this  transformation  is  identical  with  that  by  which 
the  general  equation  of  the  second  degree  in  space  is  referred  to  the 


—  108  — 

axes  of  the  surface.  Hence,  if  S^r,  Sy,,  and  S^r,  are  the  three  roots  of 
the  equation 

( c:-  s)  ( (-;-  >s')  ( c:— s) + 2  kj^:k-  ^.  \bi\  c:-^)-\ = o , 

they  are  the  squares  of  the  semiaxes  of  the  ellipsoid.  But  it  must 
be  observed  that  the  mass  of  the  ellipsoid,  being  the  same  with  that 
of  the  spheroid,  gives  the  equation 


Cf      Cf      Cf     /  ^ '"   \" 


The  condition  (104i4),  however,  shows  that  the  values  of  C^,  O^, 
and  C[,  in  (1048),  may  be  increased  or  decreased  by  the  same  quan- 
tit}",  without  changing  the  value  of  (lOlg)-  The  values  of  C'J,  C'J, 
and  C^^  may,  in  like  manner,  be  increased  or  decreased  by  the  same 
quantity,  which  change  will  produce  the  opj)Osite  effect  uj)on  the 
roots  of  (IO83),  until,  at  length,  they  may  satisfy  the  equation  (lOSg). 
This  common  increase  or  decrease  of  all  the  roots  of  (IO83)  corre- 
sponds to  the  performance  of  the  same  operation  upon  the  squares  of 
the  semiaxes  of  the  ellipsoid,  that  is,  to  a  change  of  the  ellipsoid, 
given  by  (IO83)  into  another  ellipsoid,  w^hich  has  the  same  foci  and 
the  required  mass.  The  change  of  mass  is,  however,  more  simply 
accomplished  by  an  increase  or  decrease  of  the  density  of  the  ellip- 
soid ;  and,  in  this  view  of  the  case,  it  is  requisite  that  the  value  of 
the  density  be  determined  by  equation  (IO89). 

206.  If  the  point  is  without  the  spheroid,  but  near  its  surface, 
it  is  generally  necessary  to  combine  the  forms  of  the  potential  given 
in  (99ic).     Thus,  with  the  notation 

J=  the  integral  for  all  directions  of  u  greater  than  r, 
I    =  the  integral  for  all  directions  of  u  less  than  r, 


—  109  — 
whence 

^J  -ip        *J  -ip        ty  Tp 

tlie  value  of  the  potential  may  be  expressed  in  the  form 
in  which 


r 

But  it  may  be  observed  that,  by  (lOOa)? 


0         0 

whence,  by  putting 


^^'  —  V^-'h^  J  PL'"-'' 


and  using  U,^  in  the  signification  of  (QOia),  the  potential  assumes  the 
form 


f-X^.-J^v*?-)) 


"""  n  \  ..  J/  4- 1  I 


207.     A  similar  investigation  may  be  extended  to  the  ellipsoid 
of§204,  andif 

S2'  is  the  value  of  12  of  (IOT24) 


—  110  — 

the  value  of  the  potential  for  a  imint  tvhich  is  near  the  siuface  of  the  spheroid 
may  assume  the  form 

si  =  n'-i.f\(v,,.-i-,:)Q.). 

208.  If  the  form  of  the  spheroid  differs  hut  little  from  an  ellipsoid 
which  has  the  same  foci  tvith  the  preceding  ellipsoid,  and  if  it  has  a  constant 
density  for  all  that  portion  for  tvhich  {>  is  greater  than  r,  a  combination  of 
tivo  homogeneous  ellipsoids  may  he  suhstituted for  the  single  ellipsoid,  hoth  of 
tvhich  have  the  same  foci,  ivhile  one  coincides  very  nearly  tvith  the  spheroid  in 
form  and  density  throughout  the  portion  exterior  to  r  ;  and  the  other,  hdng 
much  smaller,  has  the  requisite  positive  or  negative  density  to  give  the  alge- 
hraic  sum  of  the  masses  of  the  two  ellipsoids  equal  to  that  of  the  spheroid. 
The  combination  of  the  two  ellipsoids  upon  any  external  point  is 
the  same  with  that  of  the  single  ellipsoid,  and  the  larger  of  the  two 
may  be  substituted  for  it  in  the  values  of  Fin  (llOg). 

If,  in  determining  the  values  of  V  for  the  spheroid  or  the  ellip- 
soid from  ( 10922),  ^^  is  supposed,  for  every  direction  in  w^hich  the  solid 
is  contained  within  the  sphere,  of  which  radius  is  r,  not  to  refer  to 
the  surface  of  the  solid,  but  to  coincide  with  r,  the  value  of  F  van- 
ishes for  any  such  direction,  and  it  becomes  a  continuous  function, 
of  which  the  derivatives  are  discontinuous.  The  equation  (IOI22)  is 
applicable  to  such  a  function,  for  the  argument  by  which  it  was 
established  was  independent  of  this  condition.  With  this  modifica- 
tion, therefore,  the  accent  may  be  omitted  in  the  integral  sign  of 
(1092;)  or  (llOg),  and  the  limits  of  integration  extended  to  every 
possible  direction,  and  the  result  may  be  simplified  by  means  of 
(101^2). 

In  the  present  case,  in  which  Jc  is  constant,  equation  (IO922) 
becomes 

^<"  =  ,7+3  (     r'^-     )  +  ;r:r-,  {^^  -  ''\ , 


—  Ill  — 

whence 

If,  then,  it  is  assumed  that 

^p  =  u  —  r, 

/^^  =  u  —  r, 

e 

the  binomial  theorem  gives 

and  if  w  is  changed  into  —  [u  -\-  1) 

«-2^"  ^  )—         -Al\a-i)rm^~')  -'l')- 

These  values,  substituted  in  (III2),  give 

V>^—Vn  —  ^  r     ^      /    n^^  +  2)      ,  (-i)'»r(n-3+»ox   ,„     ^,1 
^      Kp       ;-'"L/'mr'»-iV/'(^^  +  3  — wO  "T"        r(«  — 2)       )\"p—"i^)\ 

1     ^    [_!_  /     A»  +  2)  ,     (-irr(n-3  +  »0X  _  -1 

"T"  7 '"  L / '« '•'" ~ '  \^(«  +  3  —  m)  ~f"  /•(«  —  2)  /  '^•^P         "P    ^  J  • 

This  value  may  be  substituted  in  (IIO3),  and  the  result  reduced 
by  means  of  (IOI22). 

209.  If  the  spheroid  is  not  very  different  from  a  sjjJiere,  and  if  the 
difference  in  form  hetiveen  it  and  the  larger  of  the  two  combined  ellipsoids  is 
so  small  that,  in  consideration  of  the  large  divisors,  the  terms  of  (III20)  mag 
he  neglected,  in  which  m  is  greater  than  3,  (III20)  i^  reduced  to 

V,n-V,,.  =  {2n-\-\)W„ 

c 
if 

W    1    7«/^2  y2\      1      Ji  /     3  ,/3\  . 


—  112  — 

and,  by  (110,o), 

But  the  value  of  the  potential,  derived  from  (IIO3),  becomes  in 
this  case  by  (III28)  and  (IOI22), 

477 

n  =  £1' -  i  ((2«  + 1)/  ( n;  §„)) 

0 

^  pj  _  4  ,^  2^^  Ti^['o  ^  n'  _  w,  ■ 
0 

—  (y . 

so  that,  in  iJiis  case,  the  form  (IOT24)  ^'^  ap'pUccible  to  ever?/  external  point. 
This  conclusion,  and  the  mode  of  investigation,  includes  Poisson's 
analysis  of  the  spheroid,  which  differs  but  little  from  a  sphere  by 
which  it  was  suggested. 

210.  The  formula  (IO724)  gives,  for  the  attraction  in  the  direc- 
tion of  the  radius  vector,  the  expression 

e  8  e 

Hence,  the  equation  is  obtained 

^  1  e  ^  '     e  3  e 

which,  by  (1082),  is  reduced  to 
or 

00 

e  3  6 

211.  If  the  spheroid  is  homogeneous,  having  the   same   density 
with  the  ellipsoid,  the  equation  (lOlgi)  gives 


—  113  — 


^p"* —  ::r]-^K^^y 


n 


+  3"^ 


Bj  assuming,  then,  the  values 


'  e  e 

1  /„_Lf!  „_1_Q\  TT-  1 


V'     w  +  3'^-^p       :p   >*'     '^'■"  — „+3^-^     :'•    >'' 

o'  __  o i2  • 

e 

the  equation  (11228)  becomes 

212.  7/^  ^/^e  attracted  point  is  upon  the  surface  of  the  spheroid,  the 
preceding  equation  becomes,  if  P.^  is  the  potential  at  the  surface  of 
the  spheroid, 

D,n.',  +  Ul',  =  -  271  Kr 3:^  v^^. 

213.  If  the  spheroid  differs  so  little  from  the  ellipsoid  that  the  square 
of  the  distance  between  the  surfaces  of  these  two  solids  may  he  ner/lectcd,  the 
notation 

e 

gives 

^p"  —  ■' p     Up  —  -^p     Up- 

214.  If,  moreover,  the  ellipsoid  differs  so  little  from  a  concentric 
sphere,  that  the  product  of  the  difference  between  the  radius  vector 
of  the  ellipsoid  and  the  radius  of  sphere,  multiplied  by  the  distance 
between  the  surfaces  of  the  ellipsoid  and  the  spheroid,  may  be  neg- 
lected, the  preceding  equation  is  reduced  to 

15 


—  114 


and  (llon)  becomes 


In  this  last  form,  the  sum  of  the  terms  in  the  second  member  is 
extended  to  include  the  whole  series,  because  the  first  terms  which 
vanish  in  the  exact  formula,  may  become  sensible  in  the  approxi- 
mate form.     But, 


V  —  ^  ;/f"^  • 

0 


and,  therefore,  if  II  is  the  radius  of  the  sphere, 

Ai2J  +  ^,.Q;  =  -2.rA^i?y,. 

215.     If,  again, 

£1^  =  the  potential  of  the  ellipsoid  at  its  surface, 

e 

S2q  =z  the  potential  of  the  ellipsoid  at  the  surface  of  the 

e 

spheroid, 

e  e  e 

the  general  equations 

e  0  « 

give 

Since  the  second  members  of  these  equations  are  midtiplied  by 


—  115  — 

y^,  the  values  of  the  other  factors  may  be  reduced  to  those  which 
belong  to  the  sphere.  Hence,  ^p«  becomes  a  constant  quantity, 
and,  therefore, 

for  all  values  of  n  except  zero,  in  which  case, 

e  "  e 

and  the  above  values  become 

e 

which  give 

The  sum  of  this  equation  and  (114i2)  is,  by  (llog)  and  (1142o), 

216.     If  fhc  dlijisoidis  itself  the  sphere,  the  equation  (SSg)  gives 

e 
e 

T)  r)     I   _L  o^  — 2  jiXR ' 

which,  substituted  in  (lloi;),  gives 

T)  n  _i_J_r)  — i-r KR 

This  is  the  equation  given  by  Laplace  for  a  spheroid  which 
differs  but  little  from  a  sphere,  and  is  the  fundamental  theorem  of 
his  investigations  upon  this  subject. 


—  116  — 

217.  If  the  aitr acted  foint  is  ivitJiin  the  spheroid,  and  at  such  a  dis- 
tance from  the  surface  that  r  is  less  than  the  value  of  u,  the  formula  for 
the  potential  is,  by  §  195. 

in  which 

4  TT  »■ 


0     '  0 

V'=f''  f^. 

J ih  Jp  Q 


0    ' 


It  may  also  be  shown  by  the  method  of  §§  208-209,  that  this 
same  formula  is  applicable,  if  tlie  point  is  quite  near  the  surface,  and  if  the 
spheroid  differs  so  little  from  a  sphere  that  the  square  of  the  difference  may 
he  neglected. 

218.  The  important  discussions  in  regard  to  the  convergency 
of  the  series,  derived  from  Legendre's  functions,  are  deferred,  on 
account  of  their  great  length,  to  the  volumes  which  will  be  devoted 
to  the  application  of  the  Analytic  Mechanics. 


IV. 

ELASTICITY. 

219.  The  laws  by  which  the  elementary  forces  oi  cohesion  and 
affinity  vary  with  the  mutual  distance  and  direction  of  the  particles 
and  atoms  are  undetermined ;  and,  therefore,  the  delicate  inquiries 
involved  in  the  constitution  and  crystallization  of  bodies  are  not  yet 
subject  to  the  control  of  geometry.  But  it  is  sufficiently  apparent 
that  these  forces  are  insensible  at  sensible  distances,  and  that  there  are 
peculiar  laws  of  mechanical  action  corresponding  to  the  three  states 


—  117  — 

of  f/asses,  liquids,  and  solids.  The  peculiaritv  of  these  states  consists, 
prmcipally,  in  the  faciUty  with  which  the  particles  can  be  moved 
relatively  to  each  other,  and  in  the  phenomena  which  arise  from 
such  motion,  but  especially  in  those  of  the  disruption  of  solid  bodies. 
As  long,  however,  as  the  relative  positions  of  the  particles  are  so 
little  disturbed  that  they  return  to  their  initial  state  when  the  dis- 
turbing cause  is  removed,  the  precise  law  of  molecular  action  is  not 
required  for  the  investigation  of  the  small  changes  which  the  consti- 
tution of  the  body  undergoes,  and  which  are  treated  as  phenomena 
of  elasticitij. 

220.     To  analyze  the  changes  of  form  of  a  system  of  material 
points  which  constitute  a  body,  let 

u  be  the  distance  by  which  a  point,  of  which  the  coordinates 
are  .r,  y,  and  0,  is  moved  from  its  initial  position, 

A  the  increment  of  a  function  for  another  point  of  the  body 
which  is  near  the  former  point, 

IJ  the  distance  of  the  second  point  from  the  former  point ; 


the  notation  of 

(42,) 

gives 

P 
^pcos^, 

^p. 

=PxD^ 

u^  ^PyDyU^ 

-\-P-^ 

A 

?r 

Hence,  if 

P 

=P  +  ^'Jp, 

Ap 
~   P  ' 

£  is  the  linear  expansion  of  the  body  in  the  direction  of  ^; ;  and 
its  value  is  given  by  the  equation 


(i+.)-(^y=^.a 


2 

p 


—  118  — 

=  ^,[(1  +  i>,!gcos^  +  D,tt,conl  +  A«.cosi']'. 

If,  then,  the  reciprocal  of\-\-^is  laid  off  from  the  origin  upon  a  line 
draivn  parallel  to  p,  its  cxtremitf/  ivill  he  upon  the  ellipsoid,  of  ivhich  the 
equation  is 

1  =  ^.,  [(1  +  D^ur)  X  +  D.u^y  +  D^ji^iJ. 

221.  The  expansions  or  contractions  which  corresjiond  to  the 
axes  of  this  ellipsoid  may  be  called  the  p>fincipal  expansions  and  con- 
tractions, and  one  of  these  is  a  maximum,  another  is  a  minimum,  and 
the  third  is  a  maximum  in  some  directions  and  a  minimum  in  others. 

If  the  ellipsoid  is  referred  to  its  axes,  the  expression  for  the 
expansion  is,  if  e^;?  ^y?  and  e.  are  the  values  of  g  for  the  axes, 

SO  that  for  these  directions  the  values  o^u^,  Uy,  u,,  are  such  that 
(1  +  I),u,)D,^u,  +  n^Uyil  +  DyU,)  +  BjLDytL  =  0. 

(1  +  e.^y  =  (1  +  B^u^y  +  (i>,.g^  +  (i>,,gi 


222.     The  notation 


P 


iJTives 


cosf/)  =  -ZJcos^cos^  ) ; 
sin^  cp  =  1  —  cos^fj) 

==  JS-.cos^^.^^cos^^  —  [^.(cos^cos-^)] 

= -^^^cosycos^  — cosCcoSy  I  ; 


'\12 


119  — 


(1  -|-  f)^sin^f/j  =  —.^^(pcos^cos^  — ycosjcos^j 

=^  ^J  (^cos^i>^  -)-  COS y  By  -f-  coh^^dMuzCos^  —  i^yCosl^)  1 


|2 


in  wliicli  the  derivatives  are  only  applicable  to  ii^,  Uy,  and  il. 
Hence,  if  the  reciprocal  of  the  square  root  of  {\  -f-  *)  sinf/)  is  laid  off  from 
the  origin,  upon  a  line  drawn  parallel  to  p,  its  extremity  is  upon  the  surface 
of  the  fourth  degree,  of  ivhich  the  equation  is 

223.  When  the  axes  are  those  of  the  ellipsoid  of  §  221,  and 
the  disturbance  is  such  that  for  each  axis  the  equations  (llSjg)  and 
(II829)  become 

(l-|-e)^sin^9)  =  ^^|coSy  cos^(Z>,2f.  —  DyU,^ 

=  ^^[cOS^COS^(fc,  —  £^)J  . 

224.  To  determine  the  rotative  effects  of  the  disturbance 
about  the  axes,  let 

P 

and 

ip^^  =  the  projection  of  the  angle  (p  upon  a  plane  perpendicular 

to  the  direction  of  ^. 
Hence 


—  120  — 

,        /         ,  N  pi I),ti,cosP  -\-  Z),,M,cosi;  4-  (1  -f-  Am,)  cosP 

225.  If  the  axis  of;?;  is  perpendicular  tojK»j  the  equations  are 

P  —  Xfy—P 

tan  (y,  -h  1/' j  -    1  _^  ^^,^,^  ^  A^^tanx/;  ' 

226.  If  the  axes  and  conditions  are  those  of  §  223,  the  equa- 
tion (I2O3)  becomes 

tan(f/),  +  1/',)  =  \'\_'^^ll  tani/^,. 

227.  The  whole  expansion  or  contraction  of  the  body  at  any 
time,  is  derived  from  the  consideration  that,  by  the  definition  of  e  in 
§  220,  any  very  minute  portion  of  the  body  which  is  originally  a 
sphere,  becomes,  in  the  disturbed  state,  an  ellipsoid  similar  to  that 
of  §221.     If,  then, 

^  1=  the  expansion  of  the  body  ; 

the  sphere  of  which  the  radius  is  i,  becomes  an  ellipsoid,  of  which 
the  axes  are  i{l  -\-  ^^),  i{l  -\-  ty),  i{l  -^  aS),  and,  therefore,  its  vol- 
ume becomes 

and 

l  +  «  =  (l  +  e,)(l  +  g(l  +  a,). 

228.  When  the  disturbance  is  so  small  that  the  squares  of  the 
expansions  may  be  neglected,  which  is  the  ordinary  case  of  elas- 


—  121  — 
ticity,  the  equation  (119,3)  becomes 

«  =  —  .r  [cos  2  J  A?'..  +  COS  ^  COS  ^'  {D,jlL  -\-  A?^)J 

=:  :r.,  (cos  I  a)  ^..  (cos  ^!  u)j  . 

229.  If,  then,  the  reciprocal  of  the  square  root  of  the  linear  expamion 
in  any  direction  is  laid  off  from  the  origin  upon  that  direction,  as  the  radius 
vector  of  a  surface,  the  residting  surface  is  a  surface  of  the  second  degree,  of 
which  the  equation  is 

or  l=:^,(xi>J^,(x«J. 

230.  If  the  axes  are  those  of  the  principal  expansions  and 
contractions,  the  formula  for  expansion  becomes 


£  =  ^.(cos2j.fc^); 


and  the  equations  of  §  221  become 

Djly-\-     DyU^=^       0 


X  '■'^x' 


231.  If  the  principal  expansions  and  contractions  are  all  of 
the  same  name,  that  is,  if  all  are  expansions,  or  if  all  are  contrac- 
tions, the  surface  of  §  229  is  an  ellipsoid.  But,  in  other  cases,  in 
which  neither  of  the  principal  expansions  is  zero,  the  surface  is  the 
combination  of  two  hyperboloids,  of  which  one  is  one-parted,  and 
the  other  is  bi-parted.  Both  these  hyperboloids  have  the  same 
axes  and  the  same  asymptotic  conical  surface  ;  and  the  asymptotic 
conical  surface,  corresponding  to  the  directions,  in  which  there  is 
neither  expansion  nor  contraction,  divides  the  directions  in  which 
the  solid  is  expanded  from  those  in  which  it  is  contracted. 

IG 


—  122  — 

If  one  of  the  principal  expansions  is  zero,  the  surface  is  reduced 
to  a  cjh'nder ;  and  if  two  of  the  principal  expansions  are  zero,  the 
surface  is  reduced  to  two  parallel  planes. 

232-.  In  the  present  case,  the  formula  of  §  227,  for  the  expan- 
sion of  the  soHd,  is  reduced  to 

233.  The  formula  (I2O9)  for  the  rotation  about  the  axis  of  x 
becomes, 

tan  {(p,  +  i/')  =  t'-in  if'  +  -^ 

=  (1  -|-  Z>-?f,  —  J^yif,/)  tani/^  —  D^u,,iaii^  i\i  -\-  D^iL, 
(p..  =  li^^y^'z — A?^)  +  li^y^iz  +  A«y)cos  2 1/^  -|-  1{D^  11^ — Dy  iiy)  sin  2 1^ 
=  n.  +  "^x  cos  2  ( J/;  —  1],)  ; 

in  which 

T^  sin  2  ij^  =  1  (A  ?fz  —  ^y  "y)  • 

234.  The  maximum  rotation  about  a:  corresponds,  then,  to 

W  —  ''U 
and  is  c/)^  =  i/,  -[-  t^  ; 

and  the  minimum  rotation  corresponds  to 

r  =  Vx  +  ^, 

is 

and  Jf^  is  the  mean  rotation.      When  the  maximum  and  minimum 
rotations  have  opposite  signs,  there  are  two  intermediate  rotations 


—  123  — 

which  vanish,  corresponding  to 

cos2(i/;  — ry,)=— ^. 

235.  There  are  simiLar  fonniilre  for  rotations  above  the  axes 
of  y  and  z,  and  the  combinations  of  the  mean  rotations  give  a  great- 
est mean  rotation,  represented  by 

the  direction  of  which  is  determined  by  the  equations  represented 

by 

Tl        IT, 

cos,=-. 

236.  If  the  axes  are  those  of  §  230,  the  equations  of  §  233 
become 

f;,  =  77-,  +  M  A  v..  —  D,  li,)  sin  2  w . 

237.  When  the  disturbance  is  such  that,  for  each  of  the  prin- 
cipal axes,  there  is  the  equation 

the  equations  of  the  preceding  section  become 

77,  =  77=0, 

9,  =  1  (Z>,«,  —  DyU^)  sin  2  w  ; 

so  that,  in  this  case,  there  is  compression  vjithout  any  mean  rotation. 

238.  When  the  disturbance  is  such  that  for  each  of  the  princi- 
pal axes 


—  124  — 

the  equations  for  compression  and  rotation  becomes 

so  that,  in  this  case,  there  is  rofation  ivitJiout  compression. 

All  the  preceding  investigations  upon  the  internal  changes  pro- 
duced by  the  disturbance  of  the  form  of  a  body  are  taken  from 
Cauchy. 

239.  The  elastic  force  which  is  developed  by  any  small  dis- 
turbance of  the  internal  condition  of  a  body  is  proportional  to  the 
amount  of  disturbance,  and  has,  therefore,  the  same  general  form 
with  that  of  the  disturbance  itself  But  the  special  discussion  of  the 
relative  values  of  the  coefficients  involves  the  consideration  of  the 
laws  of  equilibrium,  and  must  be  reserved  to  a  subsequent  chapter. 


V. 

MODIFYING    FORCES. 

240.  Among  the  forces  of  nature,  those  which  produce  the 
equations  of  condition  deserve  peculiar  consideration.  Being 
merely  conditional,  they  do  not  augment  or  decrease  the  power 
of  a  system,  but  merely  modify  its  direction  and  distribution.  They 
may,  therefore,  be  called  modifying  forces  ;  and  may  be  divided  into 
two  classes  of  siaiionary  and  moving. 

241.  Stationary  modifying  forces  are  perpendicular  to  fixed  sur- 
faces or  lines,  and  constitute  the  action  by  which  certain  material 
points  of  a  system  are  restrained  to  move  upon  those  surfaces  or 
lines.     A  force  of  this  nature,  being  perpendicular  to  the  motion  of 


—  125  — 

its  point    of  application,   does  not  increase    or  diminish  the    total 
power  of  the  system,  but  modifies  its  elements  of  direction. 
Thus  the  equation  of  condition, 

X  =  0, 

between  the  coordinates  of  a  point,  involves  the  idea  of  a  force, 
acting  in  the  direction  i\'^of  a  normal  to  the  surface  represented  by 
this  equation.  When  it  is  combined  with  its  multiplier,  it  is  equiva- 
lent, by  (STie),  (^Ts),  and  (543i),  to  a  modifying  force,  of  which  the 
magnitude  is 


"o 


242.     This  force  may  be  decomposed  into  three  forces,  wdiich 
are  parallel  to  three  rectangular  axes,  either  of  which  is  represented 

by 

?.v'(aX)cosf, 

while  the  point  of  application  moves  through  the  elementary  arc 
ds,  its  advance  in  the  direction  of  the  axis  of  x  is 

^5008*. 

The  amount  of  power  added  to  the  system,  by  the  component 
force  in  the  direction  of  the  axis  of  .r,  is 

Xt/5v/(n-^)cosfcos*, 

and  there  is  a  consequent  increase  or  diminution  of  force  in  this 
direction.  But  the  mutual  perpendicularity  of  iVand  s  is  expressed 
by  the  equation 

-Z^(cosfcos^)  =  0. 

The  whole  augmentation  of  power  arising  from  the  three  com- 
ponents is,  therefore, 

>.^5v/(n^)-^.(cos^cos^)  =  0, 


—  126  — 

which  agrees  with  the  fundamental  conception  of  a  stationary 
modifying  force,  and  illustrates  its  mode  of  action. 

243.  Moving  modifying  forces  are  ^^ei'pendicular  to  moving  sur- 
faces, which  surfaces  are  themselves  portions  of  the  moving  system, 
and  the  points  of  application  are  restrained  to  move  upon  these 
surfaces.  In  this  case,  the  motion  of  each  point  of  application  may 
be  decomposed  into  two  parts,  of  which  one  part  is  perpendicular, 
and  the  other  is  parallel  to  the  moving  surfaces.  The  modifying 
force  has  the  same  relation  to  the  motion  which  is  perpendicular  to 
it,  which  has  been  already  discussed  in  reference  to  the  stationary 
surface  ;  put  by  its  relation  to  the  other  component  of  the  motion, 
it  communicates  power  to  the  point  of  application,  or  the  reverse. 
But  the  power  which  is  thus  communicated  to  the  point  is 
abstracted  from  the  surface,  and  through  it  from  the  other  por- 
tions of  the  system ;  and,  therefore,  the  whole  amount  of  power  of 
the  system  is  neither  increased  or  decreased.  Although  for  the  pur- 
poses of  theoretical  speculation,  it  is  convenient  to  regard  the  sur- 
face and  the  point  of  application  as  parts  of  one  system,  it  is  often 
the  case  in  the  useful  arts  that  this  transfer  of  power  is  of  the 
highest  practical  importance,  and  is  the  basis  of  the  theory  of  the 
turbine  wheel. 

In  a  rigid  system  of  bodies,  these  forces  constitute  the  honcls  of 
union. 


—  127  — 


CHAPTER   VI. 
EQUILIBRIUM  OF   TRANSLATION. 

244.  The  conditions  io  ivliich  amj  comhinaiion  of  forces  must  he  sub- 
ject, in  order  tlicij  may  not  tend  to  produce  translation  in  the  system  of 
material  points  to  ivhich  they  are  applied,  are  readily  investigated.  It 
follows  immediately  from  §§  18  and  20,  and  with  the  notation  of 
those  sections,  that  the  algebraic  condition  that  the  system  has  no 
tendency  to  move  in  the  direction  of  ^;  is 

^;w2il\cos/=:0. 
But  each  term 

«?l^lCOSj  , 

is  the  projection  of  the  force  m^F-^  upon  the  direction  of  p,  and, 
therefore,  if  the  algehrcdc  sum  of  the  projections  of  all  the  forces  upon  any 
direction  vanishes,  there  is  no  tendency  to  translation  in  that  direction. 

245.  It  also  follows  from  the  combination  of  translations, 
given  in  §  23,  that  if  there  is  no  tendency  to  translation  in  two  different 
directions,  luhich  are  not  pfarallel,  there  is  no  tendency  to  translation  in  the 
plane  of  these  two  directions ;  and  if  there  is  no  tendency  to  translation  in 
three  directions,  tvhich  are  not  in  the  same  place,  there  is  no  tendency  to 
translation  in  any  direction. 

By  means  of  rectangular  axes  the  algebraic  conditions,  which 
are  necessary  and  sufficient  to  produce  equilibrium  in  respect  to 
translation,  are  combined  in  the  formula 

^.[-;(;;^i/^icos;)]2=0. 

This  formula  is  independent  of  the  situation  of  the  points  of  the 


—  128  — 

system,  except  so  far  as  the  elements  of  position  are  implicitly  con- 
tained in  the  expressions  of  the  forces  and  their  directions ;  it  would 
remain  unchanged,  therefore,  if  all  the  points  were  condensed  into 
one,  without  any  variation  of  the  magnitude  and  direction  of  the 
forces.  The  conditions  of  equilibrium  are,  then,  the  same  as  if  all 
the  forces  were  applied  at  a  single  point. 

246.  If  one  of  the  points  of  the  system  were  subject  to  the 
condition  of  being  confined  to  a  fixed  surface  or  line,  the  conditions 
of  equilibrium  of  translation  w^ould  simply  be  reduced  to  the  condi- 
tion that  the  7'esultant  of  all  the  other  forces  umild  he  ijerpendiciilar  to  this 
surface  or  line,  and  the  modifying  force  hj  tvhich  the  ijoint  ivas  restrained 
ivoiUd  he  equal  and  opposite  to  this  residtant. 

If  a  point  of  the  system  was  absolutely  fixed,  or  if  three  differ- 
ent points  were  restrained  to  move  upon  three  fixed  surfaces,  there 
would,  in  general,  he  no  possihility  of  translation,  hut  the  residtant  of  all  the 
forces  applied  to  the  system  ivould  he  equal  and  opposite  to  that  of  the  modi- 
fying forces  hy  which  the  points  ivere  confined. 

247.  The  theory  of  the  equilibrium  of  a  point  is  wholly 
included  in  that  of  its  translation.  But  since  every  system  is  a 
mere  combination  of  points,  the  complete  theory  of  equilibrium  can 
easily  be  evolved  from  that  of  translation.  This  mode,  however,  of 
arriving  at  the  conditions  of  equilibrium  is  neither  luminous  nor 
instructive. 

248.  The  conditions  of  the  equilibrium  of  translation  of  a  sys- 
tem, which  is  free  from  the  action  of  all  stationary  modifying  forces, 
may  assume  the  form,  that  each  force  is  equal  and  oppodte  to  the  result- 
ant of  all  the  other  forces. 

If,  then,  there  are  only  two  forces,  they  must  be  equal  and  opj^o- 
site ;  and  if  there  are  three  forces,  they  must  all  lie  in  the  same 
plane,  and  be  represented  by  the  sides  of  a  triangle  formed  by  three 
lines  which  have  the  same  directions  with  the  forces ;  so  that  each 


—  129  —  ' 

force  mmt  he  proportional  to  the  sine  of  the  angle  included  hetiveen  the  other 
two  forces.  Whatever  are  the  forces,  if  we  were  to  start  from  a 
point,  and  proceed  in  the  direction  of  either  of  the  forces,  through  a 
distance  proportional  to  the  intensity  of  that  force,  and  proceed 
again,  in  the  same  way,  from  the  point  at  wdiich  we  arrived  in  tiie 
direction  of  another  force ;  and  so  on,  proceeding  successively  from 
each  new  station  in  the  direction  of  the  next  force,  through  a 
distance  proportional  to  that  force,  the  course  w^ould  finally  termi- 
nate at  the  original  point  of  its  commencement. 


CHAPTER  YIL 
EQUILIBRIUM  OF   ROTATION. 

249.  The  conditions  to  ivhich  a  system  of  forces  must  he  siihject,  in 
order  that  it  may  not  tend  to  produce  rotation  about  a  point  or  an  axis,  are 
directly  deduced  from  §§  84  and  88,  and  are  simply,  that  the  resultant 
moment  of  all  the  forces,  tvith  reference  to  the  point  or  the  projection  of  this 
resultant  moment  upon  the  axis,  must  vanish. 

250.  When  there  is  an  equilibrium  of  rotation  about  a  point, 
the  resultant  of  the  forces  may  not  vanish,  in  which  case  there  is 
not  an  equiUbrium  of  translation.  About  any  other  point,  there- 
fore, which  is  not  situated  in  the  line  drawn  parallel  to  the  resultant 
through  this  point,  there  is  not,  by  §  100,  an  equilibrium  of  rota- 
tion ;  although  there  is  an  equilibrium  of  rotation  about  every  point  of  that 
line.     In  order,  then,  that  there  may  he  an  equilibrium  of  rotation  about  all 

17 


—  130  — 

points  of  sj)ace,  or  even  about  three  points  not  in  the  same  straight  line,  there 
mnst  he  an  equilibrium  of  translation  as  tvell  as  of  rotation. 

2-51.  In  the  same  way,  it  apj)ears,  that  if  there  is  an  equilibrium  of 
rotation  about  parallel  axes  lying  in  the  same  plane,  there  is  an  ecquilibrium 
of  translation  in  the  direction  perpendicular  to  the  plane  ;  and  if  there  is 
equilibrium  of  rotation  cd)out  parallel  axes  tvhich  are  not  in  the  same  plane, 
there  is  an  equilibrimn  of  translation  in  everg  dircciion  except  that  of  the 
parallel  axes. 

252.  If  there  is  a  fixed  point  in  a  system,  it  is  necessary  and 
sufficient  for  the  eqidlibrimn  of  rotcdion  that  the  resxdtant  moment  for  this 
point  should  be  nothing  ;  and,  in  this  case,  the  resxdtant  moment  vanishes  for 
every  point  of  the  strcdght  line  tvhich  is  drawn  through  the  fixed  point  par- 
allel to  the  resultant,  and  cdso  for  every  axis  tvhich  is  in  the  same  plane  tvith 
this  strcdght  line. 

253.  If  there  are  two  fixed  points  in  a  system,  it  is  necessary 
and  sufficient  for  the  cqidlibrium  of  rotation  thcd  the  moment  of  the  forces 
shoidd  vanish  for  the  line  2vhich  joins  the  two  points. 

254.  If  all  the  forces  are  parallel  and  equal,  there  is,  by  §  99, 
combined  with  §  250,  a  line  parallel  to  the  common  direction  of  the 
forces  for  which  the  resultant  moment  vanishes.  If  the  common 
direction  of  the  forces  is  assumed  for  that  of  the  axis  of  z,  the 
moment  of  the  force  acting  upon  a  particle  dm,  with  reference  to  an 
axis  drawn  parallel  to  that  of  ^  at  the  distance  a,  from  the  plane  of 
yz,  is 

[x  —  a)Fdm, 

and  the  whole  moment  of  the  system  is 

r   {x  —  a)F=Ff  (x  —  a). 
The  condition  therefore  that  the  moment  vanishes  for  this  axis  is 


/     [x  —  «)  =  0  ; 


—  isl- 
and the  plane  which  is  thus  drawn  at  the  distance  a  from  the 
plane  of  yz,  includes,  by  §  127,  the  centre  of  gravity.  Hence, 
the  axis,  for  ivMcli  the  resultant  moment  of  the  ixirallel,  and  equal  forces 
acting  upon  a  system  vanishes,  passes  through  the  centre  of  gravity  ;  and  if 
the  system  has  an  equilibrium  of  rotation,  and  if  there  is  afxcdpoint  in  it, 
the  centre  of  gravity  must  he  in  the  straigM  line  tvhich  is  draivn  through  the 
fixed  point  in  the  common  direction  of  the  forces  ;  or,  if  there  is  a  fixed  axis, 
the  centre  of  gravity  must  lie  in  the  plane  ivhich  includes  this  axis  and  the 
direction  of  the  forces.  It  is  also  apparent  that,  if  the  centre  of  gravity  is 
advanced  heyond  the  fixed  point  or  axis  in  the  direction  of  the  forces,  the 
equilibrium  is  stable  ;  but  if  the  centre  of  gravity  is  not  so  far  advanced  as 
the  fixed  point  or  axis,  the  equilibrium  is  unstable. 

The  ordinary  case  of  gravitation  at  the  surface  of  the  earth,  in 
which  its  variation  in  intensity  and  deviation  from  parallelism  is 
insensible  for  the  small  system  of  bodies  discussed  in  the  usual 
investigations  of  mechanics,  is  the  flimiliar  type  of  this  s^oecies  of 
force. 

255.  In  the  motions  of  translation  and  rotation  there  is  no 
motion  of  the  parts  of  the  system  among  themselves.  There  is  no 
change,  therefore,  in  the  mutual  distance  of  the  origin  and  point  of 
application  of  each  of  the  forces  which  arise  from  the  action  of  the 
parts  of  the  system  upon  each  other.  The  origin,  regarded  as  a 
point  of  application  of  the  same  force,  acting  in  the  opposite  direc- 
tion, moves  just  as  far  in  the  direction  of  the  force  as  the  actual 
point  of  application  ;  so  that  such  a  force  acts  precisely  as  a  moving, 
modifying  force,  and  has  no  tendency  to  affect  the  equilibrium  of 
translation  or  rotation.  All  the  forces,  therefore,  between  the  different 
parts  of  the  system  may  be  neglected  in  determining  the  conditions  of  the  equi- 
librium of  translation  or  rotation. 

This  mutual  relation  of  the  origin  and  point  of  application  of 
the  force,  by  wdiich  either  may  be  regarded,  at  pleasure,  as  beirtg 


—  132  — 

the  origin  or  the  point  of  apjDlication,  by  a,  simple  reversal  of  the 
direction  of  the  force  without  any  change  of  its  intensity,  is  com- 
monly expressed  by  the  proposition  that  action  and  reaction  are  equal. 


CHAPTER   VIII. 
EQUILIBRIUM  OF  EQUAL  AND  PARALLEL  FORCES. 


MAXIMA   AND    MINIMA    OF    THE    POTENTIAL. 

256.  In  order  to  give  precision  to  the  modes  of  expression, 
and  have  the  benefit  of  well-known  terms  and  forms  of  speech,  the 
force  considered  in  this  chapter,  is  assumed  to  be  the  typical  force 
of  gravitation  at  the  surface  of  the  earth,  acting  within  a  space  small 
enough  to  admit  of  the  neglect  of  its  variation  of  intensity  and  deviation 
from  parallelism. 

The  level  surfaces  of  this  force  are  horizontal  planes,  and  the  potential 
decreases  uniformlg  ivith  the  increase  of  height  ahove  the  earth's  surface. 

257.  Let  the  three  rectangular  axes  be  so  assumed  that  the 
plane  of  xz  is  horizontal,  the  axis  of  g,  the  upward  vertical,  that  of 
X,  the  northern  horizontal  line,  and  that  of  z,  the  western  horizontal 
line.     If,  then, 

g  is  the  intensity  of  the  force  of  gravity, 
y       G  the  distance  of  the  centre  of  gravity  from  the  origin,  and 


—  133  — 

£2q  the  value  which  the  potential  would  assume,  if  all  the  points 
were  in  the  plane  of  .-r.*  ; 

the  actual  value  of  the  potential  is,  1jy  the  property  of  the  centre  of 
gravity, 

=  £2,—  f  G^  =  S2,  —  mG^. 

Hence  the  potential  is  a  maximum,  ivhen  the  height  of  the  centre  of 
gravitij  is  a  minimum,  and  such  a  position  of  the  system  corresponds,  hj  §  62, 
to  that  of  stable  equilibrium  ;  hut  the  'potential  is  a  minimum,  when  the  height 
of  the  centre  of  gravity  is  a  maximum,  and  such  a  position  corresponds  to 
that  of  unstahle  equilihriwn. 

258.  Since  the  direction  of  gravity  is  the  same  for  all  the 
points  of  the  system,  there  cannot  he  an  equilihnum  of  translation,  unless 
there  are  stcdionary  modifying  forces,  the  resultant  of  tvhich  must  he  exactly 
cqucd  to  the  whole  weight  of  the  system,  and  have  a  verticcd,  upvydrd  direc- 
tion. 

259.  The  resultant  moment  of  all  the  forces  of  gravity  van- 
ishes for  the  centre  of  gravity ;  and,  therefore,  the  resultant  moment  of 
all  the  stationary  modifying  forces  must  vanish  for  the  same  point. 

260.  If  there  is  but  one  modifying  force  in  the  system,  it  must 
he  vertically  directed  upimrds,  have  an  intensity  equal  to  the  lohole  iveight  of 
the  system,  and  its  line  of  action  must  pass  through  the  centre  of  gravity. 

261.  If  there  are  but  two  stationary  modifying  forces,  they 
must  lie  in  a,  common  plane,  tvhich  is  vertical,  and  includes  the  centre  of 
gravity,  their  resultant  must  have  an  upivard  direction,  and  he  equal  to  the 
iveigM  of  the  system,  and  they  must  he  reciproccdly  proportional  to  the  dis- 
tances of  their  directions  from  the  centre  of  gravity.  This  last  condition  is 
involved  in  the  necessity  that  the  resultant  moment  must  vanish 
for  the  centre  of  gravity. 


—  134  — 

262.  If  the  intensity  of  the  force  of  gravity  were  to  be 
increased  or  diminished,  the  conditions  of  the  position  of  equiUb- 
rium  would  not  be  changed,  but  intensity  of  the  modifying  forces 
would  be  proportionally  increased  or  diminished.  Even  if  the  force 
of  gravity  were  to  be  made  negative,  that  is,  if  the  direction  of  its 
action  were  to  be  reversed,  the  conditions  of  the  position  of  equilib- 
rium would  still  remain  unchanged,  provided  that  the  modifying 
forces  were  of  such  a  nature  that  the  direction  of  their  action  would 
also  be  reversed  ;  but,  in  this  case,  the  position  of  stable  equilibrium 
becomes  that  of  unstable  equilibrium  and  the  o^Dposite.  This  rever- 
sal of  the  direction  of  gravity  is  relatively  accomplished  by  the 
rotation  of  the  whole  system  about  a  horizontal  axis. 


II. 

THE    FUNICULAR    AND    THE    CATENARY. 

263.  When  the  points  of  application  of  a  system  of  forces  are 
united  by  a  single  continuous  chord  which  is  destitute  of  mass,  the 
polygon,  which  is  formed  in  the  situation  of  equilibrium,  is  called  a 
funicular.  The  general  conditions  of  such  a  system  involve  a  mere 
repetition  of  the  principles  of  equilibrium  ;  and  the  present  discus- 
sion is  limited  to  the  case,  in  Avhich  the  points  of  application  are 
masses  acted  upon  by  gravity. 

264.  When  there  is  but  one  fixed  point  to  the  sj^stem  which 
may,  Avithout  any  essential  loss  of  generality,  be  assumed  to  be 
either  extremity  of  the  chord,  in  every  position  of  cquilihrimn  the  cJiord 
must  he  vertical. 

But  if  the  idea  of  the  incompressible  rod  is  supposed  to  be 
included  in  that  of  the  inextensible  chord,  each  portion  of  the  chord 
included  between  two  successive  masses  may  be  assumed  to  have  a 


—  135  — 

vertical  direction,  either  upwards  or  downwards ;  so  that,  if 

n  is  the  number  of  masses, 
2 "  is  the  number  of  positions  of  equihbrium, 

all  of  these  positions,  except  that  one  in  which  every  portion  of  the 
cord  is  directed  downwards,  involves  an  element  of  instability,  and 
must,  therefore,  be  regarded  as  abwlidely  iimtahle.  The  tension  of  each 
jyoiiion  of  the  chord  is,  in  ever?/  case,  equal  to  that  of  all  the  weight  which  it 
has  to  sustain  ;  that  is,  to  tlie  sum  of  all  the  subsequent  masses  ivhich  lie 
upon  the  fortion  of  the  chord  not  attached  to  the  point  of  suspension. 

2 Go.  When  there  are  two  fixed  points,  the  whole  included 
chord  must  hang  in  the  same  vertical  plane  w^ith  these  two  points. 
The  tensions  of  the  various  portions  of  the  chord  rejjresent  modifying 
forces ;  and  the  surfaces  at  which  these  forces  act  are  those  of 
spheres,  all  the  centres  of  which  are  movable,  except  those  of  the 
two  fixed  points.  In  the  position  of  equilibrium,  however,  all  the 
centres  become  stationary,  and  the  conditions  of  equilibrium  of  each 
mass  or  portion  of  the  chord  admit  of  independent  discussion. 

The  forces  which  act  upon  each  mass  are  gravity  and  the  ten- 
sions of  the  two  portions  of  chord  upon  each  side.  The  horizontal 
projections  of  these  two  tensions  must,  therefore,  be  equal  and  oppo- 
site in  order  to  balance  each  other ;  so  that  the  horizontal  projection  of 
the  tension  of  the  cJiord  is  invariable  throughout  its  zvhole  length,  and  equal  to 
the  horizontal  projection  of  the  sustaining  force  of  each  of  the  fixed  points. 

The  algebraic  sum  of  the  iipimrd  vetiical projections  of  the  tensions  at 
the  two  extremities  of  any  portion  of  the  cJwrd  must  be  equal  to  the  iveight  of 
all  the  intermediate  masses  in  order  to  support  them  against  the  force 
of  gravity. 

266.  These  two  conditions  are  necessary  and  sufficient  to 
produce  an  equilibrium  of  translation  in  any  portion  of  the  chord, 
and,  therefore,  of  the  whole  chord.     The  condition  of  the  eqiiihb- 


—  136  — 

rinm  of  rotation  of  each  portion  of  the  chord,  although  included  in 
the  preceding  conditions,  is  an  interesting  and  nseful  modification  of 
them. 

With  reference  to  the  centre  of  gravity  of  the  masses  of  each 
portion  of  the  chord,  the  moment  of  the  gravity  of  the  masses  is 
zero,  and  therefore  the  moment  of  the  tensions  applied  at  the 
extremities  must  also  vanish.  But  the  directions  of  these  tensions 
are  not  parallel,  and  therefore  their  lines  of  tension  produced  must 
meet  at  a  point,  at  which  both  the  tensions  may  be  regarded  as 
applied  without  affecting  their  tendency  to  produce  rotation.  At 
this  new  point  of  application  they  may  be  combined  into  a  result- 
ant, which  is  vertical,  because  the  horizontal  projections  of  the  ten- 
sions are  equal  and  opposite.  This  resultant  has  the  same  tendency 
to  produce  rotation  with  the  tensions  themselves,  and  therefore  it 
must  pass  through  the  point  for  which  this  tendency  vanishes, 
that  is,  through  the  centre  of  gravity  of  the  masses.  The  point  of 
meeting,  therefore,  of  the  lines  of  extreme  tension  of  amj  I'^ortion  of  a  chord 
is  in  the  same  vertical  ivith  the  centre  of  gravity  of  the  intermediate  masses. 

2G7.  If  the  two  extremities  of  any  portion  of  the  chord  are 
in  the  same  horizontal  line,  the  equal  horizontal  projections  of  the 
extreme  tensions  are  exactly  opposed,  and  therefore  the  moments 
of  the  vertical  projections  of  these  tensions  must  be  equal  with 
reference  to  the  centre  of  gravity.  The  vertical  iwojections  of  the 
extreme  tensions  of  any  iiortion  of  the  chord,  of  ivhich  the  extremities  are 
tijjon  the  same  horizontal  line,  are,  then,  reciprocally  proportional  to  their 
distances  from  the  vertical  draimi  through  the  centre  of  gravity  of  the  inter- 
mediate masses. 

268.  Since  the  horizontal  projection  of  the  tension  of  the 
chord  is  the  same  throughout  its  whole  extent,  no  portion  of  the 
chord  can  become  vertical.  If  any  portion  of  the  chord  is  hori- 
zontal, the  vertical  projection  of  its  tension  vanishes,  and,  therefore, 


—  137  — 

the  vertical  projection  of  the  chord  at  any  other  point  is  equal  to 
the  sum  of  the  weights  of  all  the  masses  intermediate  between  this 
point  and  the  horizontal  portion.     If  then 

T  is  the  tension  of  the  chord  at  any  point, 

and  if  the  axis  of  x  is  horizontal,  and  that  of  y  vertical,  directed 
upwards,  so  that 

T^  is  the  horizontal  projection  of  T,  and 
Ty  its  vertical  projection  ;  and  if 

s  is  the  arc  of  the  chord  at  any  point,  and 

m  the  sum  of  all  the  masses  included  between  the  point  and 
the  horizontal  portion  of  the  chord  ; 

the  following  equations  express  the  preceding  conditions : 


T^oK 

z=z 

T., 

Tcos; 

= 

T,= 

m 

tan^ 

= 

m 

The  inclination  of  the  chord  to  the  horizon,  therefore,  increases 
as  the  distance  recedes  from  the  horizontal  portion. 

If  the  chord  has  actually  no  horizontal  portion,  the  preceding- 
equations  are  still  applicable  by  assuming  for  in,  such  a  value  as 
would  be  required  to  correspond  to  the  vertical  tension  of  any  given 
portion  of  the  chord. 

269.  If,  in  proceeding  from  the  horizontal  portion  in  either 
direction,  the  chord  is  everywhere  ascending  or  descending,  its  hori- 
zontal direction  must  also  be  away  from  the  extremity  of  the  hori- 
zontal portion  to  which  it  is  attached  so  as  to  form  a  portion  of  a 
convex  polygon,  which  cannot  be  intersected  more  than  once  by 
any  vertical  line.     Such  a  position  of  the  chord  corresponds  to  that 

18 


—  138  — 

of  the  perfectly  stable  state,  or  to  that  of  the  most  unstable  state ; 
and  each  state  is  always  possible. 

If,  in  proceeding  from  the  horizontal  portion,  the  direction  of 
motion  changes  from  ascent  to  descent,  or  the  reverse,  the  horizon- 
tal direction  must  be  reversed  at  the  same  time,  and  so  that  the 
subsequent  portion  of  the  chord  will  form  an  arc  of  a  polygon  which 
will  include  the  preceding  portion  within  its  concavity,  and  the  con- 
cavities of  both  portions  will  be  turned  the  same  way. 

270.  The  difference  of  equation  (ISTis)  applied  to  two  differ- 
ent portions  of  the  chord  gives  the  following  eqviation  between  the 
intermediate  masses,  the  horizontal  tension,  and  the  directions  of 
tension  at  the  two  points, 

m.  —  m sin  (^'  —  %) 

2\  cos  ^' cos  ^ 

271.  If  the  masses  are  infinite  in  number,  and  arranged  in 
unbroken  continuity  so  as  to  form  the  chord  itself,  the  curve  is 
called  the  catenary.     In  this  case,  if 

h  is  the  weight  of  an  unit  of  length  of  the  chord,  the  mass 

of  an  element  is 
dm  =.  lids  ;  and  if 
^  =  the  radius  of  curvature, 

the   equation  (ISSjg),  applied  to  the   extremities  of  the    element, 
gives,  for  the  equation  of  the  catenary. 


^ 

— 

X 

:-^sec- 

X 

If 

A=: 

T. 

this 

equation 

becomes 

i>^s  =  ^)  =z  ^sec 


2  J 


—  139  — 

272,  If  the  chord  is  of  iinform  ihichicss  and  dcnsitf/  throughout  its 
length,  Jc  and  A  are  constant,  and  the  integral  of  (13 831)  is 

s  =  iltan^, 

to  which  no  constant  is  added,  because  the  arc  is  supposed  to  be 
measured  from  the  point  at  which  it  is  horizontal. 

273.  The  curve  of  the  uniform  chord  is  easily  referred  to  rec- 
tangular coordinates,  for  the  equations 


D^^g  =1 D^  s  sin  ^^=  A  sin  -^  sec 
Dt  X  =  Bi  s  cos  ^  =  ^  sec  ^ : 


2  5 

a:? 


give,  by  integration,  and  determining  the  constants,  so  that  the  ori- 
gin may  be  at  the  point  of  horizontality, 

^  =  ^(gec^  — 1), 

X  :=  J.logtan i (|- 71  —  ^) . 

These  equations  give,  by  elimination  and  the  use  of  the  nota- 
tion of  potential  functions, 

Sin^  =  tanJ  =  -^, 

274.     The  vertical  tension  of  the  uniform  chord  is 

and  the  whole  tension  is 

!r=  T^sec^=T^CoB^=T,(l_  +  l)  =  T^^{. 


—  140  — 

275.  If  the  chord  ivcre  required  to  he  of  such  a  variable  thichicss  as 
to  assume  a  given  form  of  curve,  the  law  of  this  variable  thickness  is 
given  by  the  equation 

The  vertical  tension  is 

-^  V  ^^^  ^  "^  ^^^  2T  J 

and  the  whole  tension  is 

5^=I;sec^ 

276.  If  the  thickness  of  the  chord  loere  required  to  le  ijroportional  to 

its  tension,  so  that 

T 

the  following  equations  are  successively  obtained  by  easy  transfor- 
mations 

Sin  -^,  =.  tan  '  =  tan  ^., 

J  =  log  sec  I  =:  log  Cos  ^, 
^)  =:  ^sec -^  =  ^ Cos -g  =  c^, 
rr=  T.sec^=  7;sec J=.  T,Qos^=T,cy. 

277.  i^  //^e  thichiess  he  such  as  to  give  an  uniform  horizontal  distri- 

hution  of  the  weight,  that  is,  such  a  distribution  that  the  weight  of  each 

portion  of  the  chord  is  proportional  to  its  horizontal  projection,  the 

equations  are 

^_  y,.cos^ 


—  141  — 


D^s  =  ()  =  Cscc\, 


and  the  curve  is  a  parabola,  of  which  the  transverse  axis  is  vertical. 
278.  If  the  chord  were  compressible  and  extensible,  it  would 
be  compelled  to  assume  that  thickness,  in  which  it  would  have  the 
requisite  tension ;  and  the  form  of  the  curve  would,  with  this  condi- 
tion, be  the  same  as  if  it  were  incompressible  and  inextensible. 
Thus,  if  F  denotes  the  function  which  expresses  the  given  law  of 
the  relation  of  the  thickness  to  the  tension,  so  that 

rp    -^     /7T   ? 

■'-  X  ^  X 

the  form  of  the  curve  is  given  by  the  equations 

1 


D'^.8  =  Q 


cos'^^F{sec'^,;) 


n  sin 


Dix  — 


1 

cos  J  i'' (sec  \*)' 


279.  If  the  chord  or  any  portion  of  it  is  confined  to  a  given 
surface,  the  resultant  of  gravity  and  the  tension  of  the  chord  on 
each  point  must  be  normal  to  the  surface,  and  is  balanced  by  the 
modifying  force  by  which  the  point  is  fixed  to  the  surface. 

If,  then,  the  tangent  plane  to  the  surface  is,  at  each  point, 
assumed  as  the  plane  of  x  ij  ;  if  the  axis  of  x'  is  horizontal,  and  that 
of  y  directed  upwards,  and  if 

(^/  is  the  radius  of  curvature,  at  this  point,  of  the  projection  of 
the  chord  upon  this  plane  ; 

the  curve  and  tension  may  be  determined  by  means  of  the  equa- 


tions 


142  — 


T  T^. 


^  ^- silly,  cos  J,',        ^'cos^^-cos^/' 

Z-'sin*, cos^/COSy',         ^         ^  ' 
D,T^=zJc  cos  Ir  COS  I,  =  k  COS  l^k  D,?j , 

280.  The  pressure  upon  the  surface  is  determined  by  the  con- 
sideration that  it  must  exactly  balance  the  tendency  of  each  point 
of  the  chord  to  move  in  the  direction  of  the  normal  to  the  surface. 
But  the  tendency  of  the  tension  to  move  any  point  of  the  chord  in 
any  direction,  as  that  of  ^;,  is 

In  the  case  of  the  direction  JV  of  the  normal  to  the  surface,  this 
expression  becomes,  because  s  is  perpendicular  to  JV, 


in  which 


q"  is  the  radius  of  curvature  of  the  projection  of  the  chord 
upon  the  common  plane  of  the  normal  to  the  surface,  and 
the  tangent  the  chord. 

Hence  the  pressure  sustained  by  the  surface  in  the  direction  of 
the  normal  is 

i?  =  ^^+^"COSl. 

281.     If  the  chord  is  destitute  of  imght  iqwn  any  portion  of  the  siir- 


—  143  — 

face,  {)'  hecomcs  infinite,  and  the  curve  is  that  of  the  shortest  line  tvhlch  can 
he  drawn  tipon  the  smface. 

The  tension,  in  this  case,  is  constiint,  and  the  pressure  upon  the 

surface  becomes 

T 
Q 

282.  In  the  case  of  a  cylinder,  of  ivhieh  the  axis  is  vertical,  the 
equations  become 

y' —  0 

y   ^^7 

T 

so  that  the  curve  is  the  same  when  it  is  developed  ivith  the  cylinder  into  a 
plane,  ivJiich  it  assumes  ivhen  it  hangs  freely. 

283.  In  the  case  of  a  surface  of  revolution  about  a  vertical  axis  and 
a  chord  of  uniform  thicJcness,  the  equations  become 

T=Jc{y+y,), 

^  sin  ^,  cos  ^,' 

in  which  the  angle  which  y  makes  with  y'  is  determined  by  the 
meridian  curve  of  the  given  surface,  the  plane  oixz  passes  through 
the  lowest  point  of  the  curve,  and  y^  is  the  length  of  the  chord 
which  is  equal  in  weight  to  the  tension  at  the  lowest  point. 

284.  A  special  solution  of  the  preceding  problem  is  given  by 
the  equations 

^  =  0,     lr  =  }^n, 

^  cos  'Ij, 

The  curve  is  the  circumference  of  the  circle  formed  hy  the  intersection 
of  a  horizontal  plane  ivith  the  surface  of  revolution.     The  tension  of  the 


—  144  — 

cJiord  is  the  iveigU  of  a  length  of  the  same  chord  ivhich  is  equal  to  the  dis- 
tance of  theiolane  of  the  curve  from  the  vertex  of  the  cone  drawn,  through  the 
curve,  tangent  to  the  surface. 
285.     If 

1/^  is  the  angle  which  the  projection  of/  upon  the  pLane  of 

xz  makes  with  the  axis  oi  x,  and  if 
d\\)'  is  the  elementary  angle  which  two  successive  positions 

of  y  make  with  each  other, 

this  elementary  angle  and  the  radius  of  curvature  are  given  by  the 
equations  -- — - 

d\\)' ^  m\l,d\\^ , 

\,^  —  D^'yr  +  n.Y  =  sin^,  A^/^  —  A;^ 
=  sin^,Z>,i//  —  cosl,I)yryf  =  mrpD.ii)  —  i>2,,sin^,. 

If,  moreover, 

ic'  is  the  length  of  the  tangent  drawn  to   the  meridian 

curve  at  any  point  of  the  chord,  and 
It,  the  projection  of  z/  upon  the  axis  of^, 

the  following  equations  are  obtained, 

sin  I,  =  u' .  sin  I,  D,  \\i  =  ii  D,  Y , 

i  ==  ^'  —  i>^,sin;,  =  sin^,cos^,(-  —  i)^,  log  sin  ^,^  ; 

which  substituted  in  (143i9)  gives,  by  dividing  by  sin ^,  cos ^,,  and 
transposing, 

i>„logsin'/= -J —  . 

286*     In  the  case  of  the  right  cone,  with  the  circular  hasc,  the  sum  of 


—  145  — 

y'  and  ii  is  constant ;  if,  then, 

a  ^u  -\-y, 

the  curve  is  determined  by  the  equation 

n   1        •   .         1  1  1    I  1 


r=  —  Z)„,log.sin.;,. 
The  integral  of  this  equation  is 


sni 


""  ^*'  («'  -  «'  -  yo)       («'  +  y,^)-^  -  («'  +  yo  -  2  ^0'^ 

in  which  the  constant  is  determined,  so  that  iif  may  be  equal  to  f/ 
when  the  chord  is  perpendicular  to  «'. 

The  chord  is  also  perpendicular  to  u',  when 

and  also  when 

^''  =  ^  («'  +  ^o)  ±  i  sj  [{a  +  y',f  +  4  «>a . 

When  li  is  contained  between  a  and  ?/q,  the  expression  for  the 
sine  of  the  angle  which  the  chord  makes  with  tt  is  less  than  unity, 
so  that  the  angle  is  real.  This  angle  is  also  real  when  u^  surpasses 
the  greater  of  the  roots  of  (1452i),  or  when  it  is  algebraically  inferior 
to  the  smaller  of  those  roots ;  but  the  angle  is  not  real  when  tt'  is 
included  between  these  roots,  but  is  exterior  to  the  preceding  limits 
It'  and  ?/q.  The  curve  of  the  catenary  itpon  the  vertical  right  cone  consists, 
therefore,  of  three  distinct  portions,  of  ivhich  one  is  finite,  and  included 
hetween  two  intermediate  points,  at  tvhich  the  curve  is  perpendicular  to  the  side 

19 


—  146  — 

of  the  cone  ;  tvhile  the  other  iivo  portions,  commencing  respectively  at  the  two 
points,  tvhich  are  the  highest  and  loivcst  of  those  at  tvhich  the  curve  is  perpen- 
dicular to  the  side  of  the  cone,  extend  to  an  infinite  distance.  These  portions 
have  two  of  the  sides  of  the  cone  for  their  asymptotes,  because  the  angle 
which  s  makes  with  ii  vanishes,  when  u  is  infinite. 

287.     The  finite  portion  of  the  catenary  upon  the  vertical  right  cone 
may  be  investigated  by  adopting  the  notation 

«'  +  Vo  —  2  ?/ 

sin^2r=  0082/":?, 

sin  i         a'  —  7/f, 

sin^  =  sin /sin  cp  ; 

and  that  of  eUiptic  integrals,  of  which  the  third  form  may  be  repre- 
sented by 

0 

These  equations  give 

II  z=  \  {a  -\-y'^{\  —  sin z sec [^ sin 9) 
=  2K  +  ^o)(l  — sec/isin^), 

,   cos^j3  —  sin^i  sin ^(5 


sm 


cos 


,  cos  d y/  (cos ^d  —  2  sin 2(3)  sin i  cos  d cos  (p 


cos  '^d  —  sin  '■^  ^  cos'^0  —  sin-^  p! ' 


tanl. 


2/P 


sin-'/i' 


sin^cos^cos(Jp' 

D^ii  =z  —  J  (a'  -\- y^) sin ?sec (i cos^ , 

V*  COS'u/  COS /3  COS  0 

=  2  (^^'  -h  ^0 )  ( sec  fi  COS ^  —  tan  /?  sin  (S  sec  ^ ) , 


—  147 


s  =:  Ka  -\- ?/q)  { sec  (i  ^^ (p  —  tan  (-i  sin  (-i  7f\  (p )  ; 

r^       /  tanl,Dou'  tan.-i?\n^  tan  p' sin, J 

JJ   III  = ■ —  — —      '        '  — 

9^  u'  (1  —  sec p sin i^)  cos 6*         (1 — «sina)cos 


(1  —  sec  p  sin  f1)  cos  d         (1  —  n  sin  g  )  cos  Q 

tan  j3 sin ^ sec'?    ■    sin /tan -p sing  sec /9 

?i^sin-qp 


for  it  is  found,  by  differentiation,  that 

i>,tan^-i^:^' =  i>,tant-i^^^^^ 

•?  tiinrjf  9  cos/^ 

sin  i  sin  g  (cos  "d  —  sin '  /  cos  ^  g  ) 

(cos  '^d  -\-  sin  ^  z  cos  -  g)  cos  0 

sin  z"  tan  ^(3  sin  (]p  sec /9 
1  —  /I  ^  sin  •^  (p 

288.  The  preceding  value  of  the  angle  v^'  admits  of  geometri- 
cal expression  by  means  of  the  arc  of  the  spherical  ellipse  in  the 
form  given  by  Booth. 

A  spherical  ellipse  is  the  intersection  of  a  cone  of  the  second  degree  vAth 
a  sphere  of  tvhich  the  centre  is  the  vertex  of  the  cone.     Let 

a  and  l^)  be  the  two  principal  semiangles  of  the  cone,  of  which 

a  is  the  greater,  and 
w  the  angular  distance  of  any  point  of  the  arc  of  the  ellipse 

from  its  centre ; 

and  its  equation  is  obviously 

cot-oj  1=  ^ —  =  ^ 7, {-  - — -.- 

tan-oj         tan''*^     '     tan-p 

Adopt  the  notation 

a  =^  the  arc  of  the  spherical  ellipse, 


—  148  — 

i  =  the  angle  which  the  perpendicular  to   either  of  the  cir- 
cular sections  of  the  cone  makes  with  the  axis,  which 
perpendicular  is  called  the  cf/cUc  axis, 
t  =  the  angle  which  the  focal  of  the  cone  makes  with  the  axis, 
1]  =  the  angle  of  eccentricity  of  the  elliptic  base  of  the  cone. 

If,  then,  through  the  centre  (?  (fig.  2)  of  the  spherical  ellipse, 
the  axes  AOA'  and  BOB'  are  drawn,  and  B  joined  to  the  foci  F 
and  F\  the  sides  and  angle  of  the  spherical  triangle  B  OF,  are 

BF=a,     B0  =  (j,     OF=t, 
OBF=ij,     BFO  =  ln  —  i, 

which  are  connected  by  the  equations 

cos«  =  cos  fi  cost  =  cotijtanz, 
sin  {i  =  sin  a  cos  i  =^  cot  i;  cot  £ , 

sine  =  sin  a  sin  9]  =  tan  z  tan/:?, 
cos?]  =  cos / cos  £  =:  cot«  tauf:?, 

sin  ^  =  sin  i]  cos  (i  =z  cota  tan  t . 

Let  C  and  C  be  the  points  at  which  the  cyclic  axes  cut  the 
surface  of  the  sphere.  Draw  OP  to  any  point  of  the  ellipse,  OF 
perpendicular  to  OF,  C'// perpendicular  to  CF,  6^// perpendicular 
to  Oil,  i^'/r  perpendicular  to  OH;  take  FK  equal  to  00,  and, 
draw  Z  7)/ perpendicular  to  OA.     If,  then, 

^=zL3f,    cp=:LF3I, 
l^IIOC,     V^OCE, 

the  following  equations  are  readily  obtained, 

C0S2  =  cos  00z=.  cot /I' tan  ^, 
tan^  :=  cosztan?/  =  cos^/tan^ 

=  cos " /cos  fc  tan  (/J  =  cos/cos)jtan(/) , 


—  149  — 


sin  ^  =  sin  z'sin  cp , 


SQG^'^  z=  1  -[-  cos^/cos^)^  tan^9)  =  sec^cp  {cos^if)  -\-  cos-^cos^/jsin-cj)) 
=  sec^c/)(l  —  sm^i]sm^(p  -\-  sin^7^sin-t^) 
=  860^9  (cos^t5  —  sin^j;cos^^sin^y), 

2                    o            9,>/ii            9'i9\                   l-l-  COS  -  i  tan  -  qp 
cos^'oj  =  cos-«cos-"(l  +  cos-^tan-cp)  = . ^ ^ :r.—^^  , 

9  cos^ «  ( 1  +  cos^  ^  tan^  rr )  cos  ^  a  cos  ^  9 

1 -)- cos  "^  j^  tan -^  g)  1  —  siu  - // sin  -  go ' 

.    9  sin-«cos^(rsec^^ 

sm-oi  = ■   2     .   2"^? 

1  —  sm ''  //  sin  '^  go  ^ 

^  ^ cos  j;  cos  r  cos  ^  ^ cos^^cos^sin2« 

'l"^  cos-g)  sin''^o)(l  —  sin-/^sin -qr)  ' 

•r-w  cos  ^  «  sin  ^  /?  sin  ^  /5  sin  nr  cos  ff' 

x/    (iJ 1 -. r^ — 

'?  (1  —  sin  ^;;  sin  ^  g)  ^  sin  ft)  cos  to' 

j^       2 sin^^cos^^j/sin^j^sin^'/sin^qccos^" 

^  cos  -  (9  ( 1  —  sin  - 1]  sin "  g)  ^         ' 

•     9      T-i   ,,9  sin- J  cos-/;  cos-" 

■  "■  cos'g(l  —  sin-z/sm-qp)^ 


i>,a2^i>^cu2  +  sin2to7>^," 

sin 'j3  cos  ^7/ 
cos^(?(l  —  sin^j^sinqc)^  V  cos^qcsec-^^ 

sin  ^  |3  cos  ^  ?/  /cos  -  0  —  sin  ^  ?/  cos  -  ^  sin  -  g \ 

cos-0(l — sin-?;sin-g)-  \  cos-gosec-^  / 


sin -j3  cos ^7/  /cos-/?sin-/^-sin^(]f  cos^/? -)- sin-z/sin-zyN 


0    -)  0 

sin -p  cos-/; 

cos-6  (1  —  sin^?;sin^g)  -' 


D^a  = 


sin  |j  cos  j;  sec 


1  —  sin^z/sin^qo' 

o  =  sm[-icosi]^,{—  sin-?;,  c/))  =  ^''^||^'^'^,(—  sm''i,  f/') 
289.     In  the  particular  case  in  which 


=  iJt, 


this  equation  is,  by  (I4817),  reduced  to 

o  —  tan/^sin /:?  3'V(_  7?^,  (j;) 


—  150  — 

which  substituted  in  (I475)  gives, 

VI  ^=  o  4- ian^   ^^——. 

I  '  tan  cp 

290.  For  the  length  of  the  arc  of  the  chord  which  extends 
from  its  lowest  to  its  highest  point,  this  equation  becomes 

and  if  the  magnitude  of  this  angle  is  commensurate  with  the  total 
developed  angle  of  the  cone,  Uic  chord  returns  into  itself,  after  imssing 
around  the  cone  once,  twice,  or  several  times,  dependent  upon  the  magnitude 
of  the  angle  of  the  cone. 

291.  To  investigate  the  infinite  portions  of  the  chord,  let 

/o  and  /i  be  the  roots  of  the  equation  (1452i), 
and  the  equation  gives 

44  =  —  «^o- 
Adopt  also  the  notation  of  §  287  and 

^  sin  i  (a'  +  y^  —  2  u')         cos  p'  (a'^f,  —  2  u')  —  r,-\-r,  —  2u" 

sin^'=::  sin? sine/)', 

and  the  following  reductions  are  obtained,  by  the  substitution  of 
cosec^'  for  sin  (5, 

?f'=  l(lo-{-l[){l  —  secficosecg)'), 

/>  ,s  —  1  (r/  -J-  9/\  {^_^IlM^        sec^cosVX 
'^'■^  —  2  l^*  -1-  !/o}  {-^^r  sl^T^^^ j 

=  K^'+yo)[tan/'isin/'lseci3'  +  sec/'icos(3'+  sec/':?Z)^,(cos^'cotc/)] , 
«==  K«'+^o)(sec fill  S.f/+ tan fi sin /ig^,f/+ sec /icos^' cot 9')> 


—  151  — 


—^        , sin  ^  p  cos  /?  sin  -  g;'  sec  0'  —  sin  -  ^  sin  g'  sec  d' 

^/  ^^  ~  1  — cos-/3sin2,y/ 

tan /? sin /j  sec  0'         ,        ,,   .     ,.,         ../    i     t\    x. 
=z  — -^ — .!\  .  .,  ,  —  tan  :?  sin  j  sec  ^  +  -^n'  tan 

1  —  C0B-(Ssm-qj  '  '  '9 


[_1]CO50' 

cos  q' ' 


1//  =  tan/isin/i  5?,(—  cos^.^i^')  —  tan.isin/'i^ic/  +  tan^-ii  ^'. 

292.  The  term  of  the  prececlmg  value  of  i/'',  which  depends 
upon  elUptic  integrals  of  the  third  order,  may  be  constructed  by 
means  of  a  spherical  ellipse,  of  which  the  parameter  is  the  reciprocal 
of  that  employed  in  the  construction  of  the  similar  term  of  the  finite 
portion  of  the  chord.  The  parameter  of  the  spherical  ellipse  of 
§  287  being  sin?],  the  reciprocal  parameter  is 

sin  i  ,j 

:=  COS  p  , 

sni;; 

and  the  length  of  the  arc  of  the  corresponding  spherical  ellipse  for 
the  amplitude  9'  is 

n'=  sin/icos,)g>,(-  cos^i,  9')  =  ^i^^  3>,(_  cos^'i,  ,.'). 
This  arc  is  reduced,  in  the  case  of 

to 

o'  =  tan  (3  sin  tj^.{—  cos^  ('j,  (p') . 

293.  The  finite  portion  is  exactly  circular  when 

d  —  ij^. 
In  this  case 

2  =  0  ,       [/^  ^  J  TT  =  « , 

and  the  equations  of  the  infinite  portion  become 

?«'=  d{\  —  y/2.coseca)'), 
s  =  rt  y/  2  cot  9)' 
i//  =  V^2(i-7t  — y')  — 2tanf-ii[(v/2  — l)tan(i7r— 1^/)]. 


—  152  — 

294.  As  ^0  diminishes  from  the  vakie  a,  the  finite  portion 
becomes  more  and  more  eccentric,  until  when 

hoth  the  finite  and  the  infinite  iwrtions  degenerate  into  straight  lines,  ivhich 
are  the  sides  of  the  cone. 

295.  When  g a  is  negative,  a  and  g^  cease  to  he  the  limits  of  the  finite 
portion,  and  lecome  the  limits  of  the  infinite  portion,  ivhile  ^  and  l[  become 
the  limits  of  the  finite  portion.  But  Iq  and  l[  are  imaginarg,  if  g^  is  included 
bettveen  the  values 

/o=:(-3  +  2v/2)a', 

so  that  between  these  limits  the  finite  portion  disappears,  and  the  cJiord  con- 
sists onlg  of  the  two  infinite  portions  ;  and  at  the  limits  the  finite  portion  is 
circular. 

To  investigate  the  infinite  portions  between  the  limits,  in  ivhich  the  finite 
p)ortion  disap2)cars,  let 

tan  i'  =^  sin  i^  —  1 , 

sin  ^"  =1  sin  i  sin  cp" ; 

and  the  following  equations  are  obtained  by  simple  transformations, 

sin(:Jcosr=  y/|, 

u'=z  1  («'_|_^^)(1  —  sec  fi  sec  9")  =  |-(a — ^o)(cos/i  —  secy"), 
cos^(3'=  1  -|-  tan^«'cos^9" 

=  sec  ^  (i  —  sni''^  sm  9)  ) 
=:  sec'^i  cos^d'^; 

^^''*—  2V«  -h^oiV    cos>"cos0"  sJ'l.co^O")' 

1/  /    I      /'x/.'=ec/?         .,/         '„        sec/?c'       t/        tan/?(-jj      ,A 


—  153  — 

^         ,  V^  1  tan /? sec ^'       i      /i  i.       o         \"        n     +^  .  l-ii  *=^"''' 

D  „w  =^  —  T-^^^ — ^. — ^-T/  +  V  o  tan  li  sec  (3  —  D  „  tan  ^    '^ — -;, 

9     '  1  —  cos-p^cos-g-'    I    V-  (  V  tune 

cos  ^' cos /?  sec  0"  ■  ./        ^^         ,v/ 

^= \  .,      +  COS  i  COS /5  sec  6 

1  —  ^  sin "  (jp       '  ' 

-  i?,„tau.->.(cos/^cos^"^)  -  i>,.tan-«,^;, 
i(/  =  —  coai'coi'ii 9, (—  1 ,  (j")  +  cosj'cosj'i Sfj^^" 

^  _  cos^"cos/5^,,(—  1  ,  ^'')  +  COS^'^COSflJ^^i,/ 

r_n  tan -t' 4- COS/? COS g/^ tan -(9^^  ^ 
^''^^^         tan  r  tan 6"(  1  —  cos  /?  cos  g")  ' 

in  Avliich  the  elliptic  integral  of  the  third  form  admits  of  interpreta- 
tion by  means  of  the  arc  of  the  spherical  ellipse. 

296.     When  the  negative  of  ^o  is  equal  to  a  the  equations  may 
be  greatly  simplified  and  reduced  to  the  following  forms, 

COS/?  =:  0,     cosr=  y/o-, 

^9"^'    —  ^(2  — sin--'g.")' 


,         sin  g 


cos^^/  =1  —  2  sin 2/  =  cos2i//. 


/  9 
U 


cos  -  g-"         cos  2  Ti;'  ' 


2fAM  25  //^^  ;w/«r  equation  of  the  equilateral  hi/perhola.  In  this  case,  there- 
fore, the  curve  of  the  chord  upon  the  develojjed  cone  is  an  equilateral  hjper- 
hola  ;  this  case  was  recognized  by  Bobillier  in  an  imperfect  investi- 
gation of  the  catenary  upon  the  surface  of  the  vertical  cone  of  revo- 
lution. 

20 


—  ]M  — 

297.      When  f/ic  surface  of  revolidion  is  an  ellipsoid,  of  which  the 
equation  of  the  vertical  section  made  by  the  plane  ?jx  is, 

let  a  sphere  be  constructed  upon  the  axis  of  revolution  as  a  diame- 
ter, and  let 

g)  be  the  angle  from  the  vertical  point  of  the  sphere  to  a  point 
of  which  ?/  is  the  ordinate,  so  that 

y  ^=^  A cosf/) ,     X  =  Bsin (p , 
u  =z  A{seG(p  —  cosg-))  =  yl sin c/)  tan g) , 
I)^?/  =  —  Asin(p. 

These  equations,  substituted  in  (14499),  with  proper  regard  to 
the  different  position  of  the  origin  of  coordinates,  give 

i>^logsin^,,  =  —  ^sin9)i:>,logsin^,  =  —coUp  +  ^J^^^j^p 


sm  v^ 


^'        sin  (f  (cos  cp  -j-  31)         -|  sin  2  g)  -|-  Msinq) 


in  which  iVand  3/ are  arbitrary  constants. 

298.     The  maximum  and  minimum  of  sin ^/  are  determined  by 
the  roots  of  the  equation 

cos 2 9)  -|-  3Icos(p  =z  0. 

If  these  roots  are  (p'  and  y",  the  equation  gives 

cose// cos 9)"  -]-  2=  0? 

31=  —  2  (cos  9'  +  cos  9")  =  —  4  cos  1  ((//  +  (p")  cos  }  (c//  —  (p') 
=  sec  9'  -|-  sec  cp". 

Of  the  two  roots,  therefore,  one  is  obtuse,  while  the  other  is 


—  155  — 

acute  ;  if  one  is  contained  between  ^  n  and  fTt,  the  other  is  impos- 
sible ;  and  when  both  are  real,  one  is  confined  between  I  n  and  |  n , 
while  the  other  is  without  these  limits.  The  corresponding  mini- 
mum and  maximum  values  of  sin  I,  are 

N  T  N 

-,  and 


tan(jp'sin^(jp'  tan  qt"  sin -^  gj"  * 

Both  these,  independently  of  their  signs,  are  minimum  values, 
and  when  they  are  both  absolutely  greater  than  unity  there  is  no 
catenary ;  but  if  either  is  less  than  unity,  there  is  a  corresponding 
portion  of  the  catenary.  AVhen  both  values  are  less  than  unity,  the 
catenary  consists  of  two  separate  portions,  because  there  is  between 
^/  and  (^"  a  value  ^'"  of  ^  which  satisfies  the  equation 

cos  (/J     =  —  M, 
and  the  values  of 

/  ,//         cos-ff/ — co.s2f/         sin-n[/ 

COSO)    COSCp      =  *- ^, =^ r, 

'  '  cos(j&  cosg, 

•      2      " 

cosf/)    —  C0S9   = ^^-V  =  ism^9  cosf/' , 

are  positive. 

299.     The  csj^ecial  case  of 

gives 

9^^  =  171, 

31  =z  0  ; 

sm; 


'  ^  sin  2  (]p  •* 

and  each  of  the  minimum  values  of  sin  ^,  is 

2N, 


—  15G  — 

which,  behig  less  than  unity,  may  be  expressed  by 

2i\^=:sin2«. 
This  equation  gives 

.     ,  sin  2  a 

■'  sin  z  fjf 

If,  then,  \  is  determined  by  the  condition 

o  ,  cos  2  (B 

C0S2/w  ^   r,-, 

COS  z  « 

simple  reductions  give 

5         y/(cos^2«  —  cos'^2fjf)  cos 2 « sin 2 P. 

COS^/^::^  -.    -^  :      ;  , 

y  Sin  2  (jp  sin  2  r;       ' 

+.,,.   _tan2« 


sin  2 ;. ' 

2  «  si 
sin  2  g) 


T-v  cos  2  «  sin  2  P. 

A/i^  =        .;n9^        ==  COS,,, 


1^  =  y/(sin>  +  -pcos^f/.^sec;,, 

=  v/[Kl+?)  +  i(?-l)^««2«cos2x]; 


-p.        D^s?k\^\. sin2«Z)fflS 

^  '  i3fsin  9  ^siu  ^  sin  2  qp ' 

7-v  sin  2  «  Z)-  * 

"-  '  xjsin  q  sin2  (p 

In  the  case  of  the  iwolate  ellipsoid,  the  notation 

.    o.                  2(i?2_^2)cos2« 
sin"?  zzzi  - - 

i?^  +  ^^+(i^^  — ^l-^)cos2, 

siu^  =r  sin?' sin  P., 
gives  the  equation 

B  =z  v/(i?2cos2«  +  yl^sin^ft)  $.  I . 


—  157  — 

In  the  case  of  the  oblate  elUjJSoid,  the  notation 

>/  =  i7i  — ;., 

.    ov  2  (.4-^  — ^2)  COS  2  « 


sint5'=  sinrsin^.', 
gives  the  equation 

s=:  y'(^2cos^«  -{-  B'^^m'^a)%^l'. 
In  the  case  of  the  sphere  the  equations  become 

s  =  Al; 

and  this  result  of  this  case  is  obtained  by  Bobillier.     This  case  also 
gives  the  equation 

7-x  sin  2  a 

'-  '  sin  q,  sin  2  qp 

sin  2  « 

(sin^a-|-  cos  2  a  sin  ^  A)  y/ (cos  ^  a  —  cos  2  cc  sin  "^  A)' 

which  by  the  notation 

cosr'=  tana, 

sin  6''  =  sin  i'^  sin  X , 
becomes 

^ 2sec^^^ 

^'^^  ~~  sin«(l+tan2i"sin2;.) ' 

'  since  ^  '     ■' 

=  2sin«tan^a^i//( — sec-«sin^r,  I)  -|-  2sin«9^,y,A 

,    ^,       r    i,sinj"tan(9"cos?. 

-[--tan^~^^ 


5111  « 


oOO.     Eeturning  to  the  general  case  of  the  ellipsoid,  let 


—  158  — 

a  and  /?  be  the  limiting  values  of  9  for  the  upper  portion  of  the 

curve,  and 
a  and  /i'  the  limiting  values  for  the  lower  portion  ;  and  let 

Hence  the  following  values  of  M  and  N  are  obtained 

N=^  \  sin  2  a  -|-  3/sin  a  ==  |  sin  2  /:?  -f-  3/sin  [^ , 
—  N^\  sin  2  a'  +  J/sin  a'  =  J-  sin  2  /f  +  J/sin  /r, 

T.  ^ COS  f  COS  2  // cos  t'  cos  2  // 

cos  //  cos  //        ' 

iV^=  tan?^  (COS"-?^  C0S2€ COS^i;COs27y) 

=  |tan^(cos2£  —  cos27j)  =  tani^(cos^£  —  cos^j;) 
=  |-tani/(cos2i/ —  cos2fc')^  tan?/(cos^'}/ —  cos^t') 
=1  tan  -)]  sin  «  sin  /5  =  —  tan  r[  sin  a  sin  {^' 

.     5  sin  ?/ sin  « sin  (3  — sin?/sin  r/sin^' 

^  sin  (f  (cos  1]  cos  gp  —  cos  e  cos  2  r^         sin  qp  (cos  //  cos  (jp  —  cos  e'  cos  2  //) 

^  Y'[ —  (cos  qp  —  cos  «)  (cos  fjp  —  cos  ,5)  (cos  q?  —  cos  ic')  (cos  qp  —  cos^')] 

'  ^  sin  g:(cosqp  -j-  J/) 

vT —  (cos^qp  —  2cos?/ cos£  cosqp  -(-  cos«  cos|3)  (cos-qp  —  2cos?/cos£'cosqp  -\-  cosa'cos^')] 

sin  qp  (cos  qpj  -|-  M) 

y/  [sin  ^  y  (cos  q:.  -f  J/)  ^  —  jyrg] 

sin  qri  (cos  qp  -|-  M) 

The  numerators  of  the  first  and  last  values  of  cos^,  give,  by 
direct  comparison, 

—  2  Jl/=  cos«  -(-  cos/i  -|-  cosa'-|-  cos/f  =  2  cos?j  cos£  -|-  ^  cos?/cose', 
whence 

cos  1]  cos  1/  cos  t'  =  ( COS  2  ij  —  cos  ^  jy )  COS  £  =:  —  siu  ^  1]  cos  e , 
cos  1^  COS  7/ cos  £  =  —  sin^jy'cost', 


—  150  — 
cos^j^cos^r/ — sin^T^sin->/=  cos(»;  +^/')cos('/  —  i^)  =  0, 

The  comparison  of  the  values  of  jV,  shows  that  the  value  of  )/ 
must  be  obtuse,  whence 

cosfc'  =  tan>^  cose, 
cos£  =  —  tan?/ cos  c'. 

301.  The  general  case  of  the  surface  of  revolution  admits  of  one 
integration,  by  denoting  by  v  the  ordinate  of  the  meridian  curve  of 
revolution,  which  gives 

this  equation,  substituted  in  (14429),  gives,  by  integration, 

sni ,,/  —         ,      - , 

in  which 

Vq  is  the  ordinate  of  the  meridian  curve  at  the  origin. 

This  form  of  the  equation  is,  however,  limited  to  the  case  in 
which  the  curve  has  a  point,  in  which  its  direction  is  horizontal. 
But  every  case  is  included  in  the  form 

M 


sm 


^'         ^'(y  +  yo)' 


in  which  J/ is  an  arbitrary  constant. 

302.  In  the  case  of  the  surf  ace  formed  ly  the  revolution  of  the  equi- 
lateral hyperlola  about  its  asymptote,  which  may  be  called  the  equilateral 
asymptotic  hyperholoid,  if  the  equation  of  the  revolving  hyperbola  is 


—  160  — 

the  equation  (159v4)  becomes 

%n\y,  —  — , 

and,  therefore,  the  inclination  of  the  curve  of  this  catenary  to  the  arc  of  the 
meridian  is  constant. 

When  J/ is  greater  than  1)"",  the  curve  is  impossible,  but  when 

the  catenary  becomes  a  horizontal  circle,  and 

303.  It  may  be  inferred  from  the  comparison  of  the  two  pre- 
ceding sections,  that,  upon  the  circle  of  intersection  of  any  surface  of  revo- 
lution ivlth  the  equilateral  asymptotic  hyperholoid  of  equation  (ISDsi),  the  arc 
of  the  catenary  of  cither  surface  makes  the  same  angle  ivith  the  meridian 
curve  of  that  surface.  Hence,  the  limiting  hoiizontal planes  of  the  catenary 
of  equation  (ISOig)  are  the  intersections  of  the  surface  of  revolution  upon 
zvhich  it  lies  ivith  the  equilateral  asymptotic  hyperholoid,  of  ivhich  the  equa- 
tion is 

The  catenary  extends  over  that  portion  of  surface  ivhich  lies  exterior  to 
the  asymptotic  hypcrloloid,  and  does  not  extend  over  that  portion  of  surface 
ivhich  is  included  ivithin  the  hyperholoid. 

304.  To  complete  the  solution  of  the  catenary  upon  the  equilateral 
asymptotic  hyperholoid,  the  equation  (ISOgi)  gives 

tan^,  =  —  D,,v^=  J—, — -2, 

whence  the  following  equations  are  obtained ; 

(^+.yo)'=^'cot^,, 


—  161 


^"yy  =  ~  2(y  +  yo)sin^^^ 


But  it  is  found  by  §  285  that 

Tj        secf^rtanlr (y -\- 1/,)  tan;, 

whence 

B-  w  z= ^-^ • 

>J'    t  9c5ti2.'/   ^rtc.v    ' 


2sin^'^,cos^. 


of  which  the  integral  is 


o^ 


305.  If  the  chord  is  not  strictly  confined  to  the  surface  so  as 
to  be  incapable  of  removal  from  it,  but  if  it  simply  lies  upon  the 
surface,  without  the  power  of  penetrating  it,  it  must  leave  the  sur- 
face whenever  the  pressure  becomes  negative,  that  is,  when  the 
sign  of  jR,  computed  by  (14229),  is  reversed.  The  points  at  which 
the  chord  leaves  the  surface  are,  therefore,  determined  by  the 
equation 


21 


—  1G2  — 


CHAPTER  IX. 

ACTION   OF   MOVING   BODIES. 

CHARACTERISTIC     FUNCTION. 

306.  Related  to  the  idea  of  the  potential,  and,  in  some 
respects  including  it,  is  that  of  the  action  of  a  system  as  proposed 
by  Maupertuis.  Every  moving  body  may  be  regarded  as  constantly 
expending  an  amount  of  action,  equivalent  to  the  power  which  its 
motion  represents,  that  is,  to  the  product  of  the  force  of  the  moving 
body  multijDlied  by  the  space  through  which  the  body  moves. 
Hence,  with  the  notation  of  Chapters  H.  and  IH.,  if  F  designates 
the  whole  action  expended  by  the  system,  the  action  expended  at 
each  instant  is 

and  the  total  expenditure  of  action  is 

The  function  V  is  called  by  Hamilton  the  characteristic  function 
of  the  moving  system,  and  he  has  resolved  the  problem  of  dynamics 
into  the  investigation  of  its  form  and  properties. 

307.  If  the  power,  with  which  a  system  is  moving  at  any 
instant,  is  denoted  by  T,  its  expression  becomes,  by  {^20), 

The  preceding  expressions  for  the  expended  action  give,  there- 
fore. 


—  163  — 
V==f2T. 

rEIXCIPLE    OF   LIVING    FORCES,    OR    LAW    OF    POWER. 

308.  If  12  denotes  that  function  which,  in  the  case  of  the  fixed 
forces  of  nature,  is  the  potential  of  the  moving  system,  its  change 
for  any  instant  is,  by  (3424)  ^''^^  §  ^8? 

Hence,  in  the  case  of  the  fixed  forces  of  nature,  if  U  is  an  arbi- 
trary constant, 

which  is  only  the  analytical  form  of  the  proposition  of  §  58,  and  is 
caWedi  tJie  principle  of  living  forces.  The  term  living  force  denotes  the 
power  of  a  system,  so  that  this  principle  may,  with  equal  propriety, 
be  called  the  laiv  of  power. 

CANONICAL    FORMS    OF    THE    DIFFERENTIAL    EQUATIONS    OF    MOTION. 

309.  The  equation  (815)  may  be  written  in  the  form 

=  D,:^i{ni-^v-^dS])  —  ^^{inii\d  DtSi) 

If,  then,  r^^,r^2,'^hi etc.,  are  assumed  to  be  the  independent 

elements  of  position  of  the  n  bodies  of  the  moving  system,  81,80,  etc., 
may  be  regarded  as  expressed  in  terms  of  these  elements,  so  that 

v=.D,8  =  2:^{D^8D,i]). 


—  164  — 
With  the  notation 

this  equation  is  resolved  into  the  equations  represented  by 

The  substitution  of  these  values  give,  if  T^j^^,  denotes  T  ex- 
pressed by  means  of  i^i,  i]^,  r(^,  rf^, etc., 

whence 

2),i2  =  (i>,Z>„,-i>,)r,,,,. 

This  expression  represents  the  elegant  forms  of  the  differential  equa- 
tions of  motion  given  hg  Lagrange  ;  but  the  mode  of  investigation  is 
adopted  from  Hamilton. 

310.  In  the  special  case,  in  which  the  independent  elements 
of  position  are  the  rectangular  coordinates,  x,  g,  z,  of  the  different 
points  of  the  system,  these  equations  become 

D^,  T^^  2-,  =  mx'  =  mDtX, 

Z^a-  i2  =  m  Dt  X  ==  m  D'\  x . 

When  the  coordinates  of  the  system  are  subject  to  conditions, 
these  equations  are  still  applicable,  provided  that  the  forces,  by 
which  the  conditions  are  maintained,  are  included  in  the  forces  of 
/2,  or  more  properly  of  d £1.  The  values  of  D^Q^  and  D^il  can  be 
obtained  from  the  given  differential  expression  of  12,  even  when 


—  165  — 

such  expression  is  incapable  of  integration  ;  for  this  form  gives 

311.  By  means  of  the  notation 

''/I?   '^i2j   ••••    etc.,   may   be    eliminated   from   the  value  of  T,  and 
Tjj^^  may   denote    the   resulting  value,  expressed  by  means  of  ?;i, 

7^2,  Wi,  W2? etc. 

Since  T  is  a  homogeneous  function  of  two  dimensions  in  respect 
to  i]\,  i]\,  etc.,  it  satisfies  the  equation 

whence 

But  the  variation  of  T,  derived  by  the  usual  method,  is 

which,  subtracted  from  the  previous  value  oi2d T,  leaves 

This  equation  is  equivalent  to  the  two  equations 

T)  T      — B  T 

and  Lagrange's  canonical  form  assumes  the  folloiving  expression  given  hy 
Hamilton, 

312.  But  /2  is,  in  the  case  of  the  fixed  forces  of  nature,  a 
function  of  i^i,  7j2,  etc.,  without  other  variables.     If,  then,  in  this  case. 


—  166  — 
tlie  preceding  equations  assume  the  simple  form 

which  are  given  by  H^vmilton,  in  which  X2  may  involve  the  time. 

VARIATIONS    OF    THE     CIIAEACTERISTIC    FUNCTION. 

313.     The  variation  of  the  characteristic  function,  taken  npon 
the  hypothesis  that  the  time  does  not  vary,  is 

But,  from  the  preceding  equations, 

the  sum  of  which  and  of  the  equation 
is 

The  variation  of  the  characteristic  function  is,  therefore, 

^F=  ^jj{codi]  —  cOqSijq)  -\-  id II, 

in  wdiich  coq  and  ijo  ^I'e  the  initial  values  of  to  and  ij .     If,  then,  V  is 
expressed  as  a  function  of  the  initial  and  final  coordinates,  ?;,  tu,  i^q? 


—  167  — 

and  ojo?  ^^^^  of  the  constant  II,  its  derivatives  are 

By  means  of  these  equations,  the  prollcm  is  resolved  hj  Hamilton  into 
the  determination  of  the  single  function  V. 

314.  In  the  case  in  which  the  independent  elements  of  posi- 
tion are  the  rectangular  coordinates,  these  equations  become 

10  =  m  X  =  m  Dt  x  ==  D^  V, 

ioq  =  w?.ro  =  niDiXQ  =  —  Dx^  V. 

315.  If  the  expression  of  the  forces  involves  the  velocities 
the  final  expression  of  dT  in  §  313  is  incomplete,  and  the  present 
mode  of  investigation  is  not  easily  and  simply  applicable  to  such 
cases,  which  is  of  less  importance,  because  these  cases  are  not,  in 
the  most  comprehensive  view  of  the  subject,  the  cases  of  nature. 


PRIXCIPLE     OF    LEAST     ACTION. 

316.  When,  in  the  case  of  the  fixed  forces  of  nature,  the  ini- 
tial and  final  positions  of  the  system  are  given  as  well  as  the  initial 
power  with  which  the  system  is  moving,  the  variation  of  the  charac- 
teristic function  vanishes,  and,  therefore,  the  function  is  generally  a 
maximum  or  a  minimum.  The  action  expended  by  the  system, 
which  is  measured  by  this  fimction,  is  also  a  maximum  or  a  mini- 
mum ;  or,  in  other  words,  the  course  by  which  the  system  is  com- 
pelled to  move  from  its  initial  to  its  final  position  through  the 
action  of  the  dynamic  laws,  is  that  in  which  the  total  expenditure 
of  action  is  a  maximum  or  a  minimum.  But  it  is  obvious  that,  in 
most  cases,  and  always,  when  the  paths  in  which  the  various  bodies 


—  168  — 

move  are  quite  short,  the  described  course  cannot  correspond  to  the 
maximum  of  expended  action ;  and,  therefore,  in  most  cases  the  sys- 
tem moves  from  Us  given  initial  to  its  given  final  position  ivith  the  least  possi- 
ble expenditure  of  action. 

Many  examples  can,  however,  be  given,  in  wliich  the  expended 
action  is,  in  some  of  its  elements,  a  maximum ;  although,  even  in 
these  cases,  the  expenditure  is  a  minimum  at  each  instant,  or  for 
any  sufficiently  short  portions  of  the  paths  of  the  bodies. 

317.  This  principle  of  least  action  was  first  deduced  by  Mauper- 
TUis,  through  an  a  priori  argument  from  the  general  attributes  of 
Deity,  which  he  thought  to  demand  the  utmost  economy  in  the  use 
of  the  powers  of  nature,  and  to  permit  no  needless  expenditure  or 
any  waste  of  action.  This  grand  proposition,  which  was  announced 
by  its  illustrious  author,  with  the  seriousness  and  reverence  of  a 
true  philosopher,  is  the  more  remarkable  that,  derived  from  purely 
metaphysical  doctrines,  and  taken  in  combination  with  the  law  of 
power  which  likewise  reposes  directly  upon  a  metaphysical  basis,  it 
leads,  at  once,  to  the  usual  form  of  the  dynamical  equations. 

318,  To  deduce  the  dynamical  equations  from  the  combina- 
tion of  the  principles  of  least  action  and  living  forces,  add  together 
the  two  variations  of  T., 

If  the  sum  is  introduced  into  the  variation  of  V,  the  result, 
reduced  by  the  condition  that  at  the  limits  of  integration, 

becomes 


—  169  — 


The  factor  of  ^t;,  in  this  expression,  must  vanish  by  the  princi- 
ples of  the  method  of  variations,  which  gives  immediately  the  gen- 
eral expression  of  Lagrange's  canonical  forms. 


PRINCIPAL    FUNCTION    AND    OTHER    SIMILAR    FUNCTIONS. 

319.     The  function  jS  determined  by  the  equation 

is  called  by  Hamilton  the  principal  fimdion,  and  its  variation  deduced 
from  that  of  Fis,  obviously, 

dS=dV—idH—IIdt 

=  .2',(w^7^  —  Wo^7]o)  —  Hdt. 

Hence,  if  S  is  regarded  as  a  function  of  t^,  t^q?  '^j  '^o?  ^tc, 

with  the  time  t,  its  derivatives  are 


The  principal  function  may,  therefore,  be  used  in  the  same  way  with  the 
characteristic  function  in  the  determination  of  the  motion  of  the  system. 

320.  Many  other  functions,  as  suggested  by  Hamilton,  can  be 
substituted  for  the  principal  and  characteristic  functions.  Thus  the 
function 

gives 

22 


170  — 


Hence, 


321.  The  introduction  of 

Q  =  W+  tll=f(^,{ri  o/)  +  //), 
gives,  in  like  manner, 

^u>Q  =  v>   ^"oQ  =  —  Vo} 

322.  Other  functions  can  be  formed  by  the  combination  of  V 
and  W,  or  8  and  Q.  The  combination  may  be  such  that  for  some 
of  the  coordinates,  the  function  shall  have  the  same  form  as  V  or  >S', 
while  for  the  remaining  coordinates  it  shall  have  the  form  of  IT  or 
Q,  and  the  function 

or 

can  be  substituted  for  V  or  8. 


171  — 


PARTIAL  DIFFERENTIAL  EQUATIONS   FOR  THE    DETERMINATION    OF    THE    CHARACTER- 
ISTIC,   PRINCIPAL,    AND    OTHER    FUNCTIONS    OF    THE    SAME    CLASP 

323.  By  substituting  in  the  equation 

for  1],  to,  etc.,  as  well  as  for  t  and  II,  their  equivalent  expressions,  as 
partial  derivatives  of  V,  S,  W,  Q,  U,  and  P,  partial  differential  equa- 
tions are  obtained,  the  integrals  of  which  give  the  values  of  these 
functions.  To  facilitate  the  expression  of  this  substitution,  T  and  12 
may  be  assumed  to  have  such  functional  significations  that 

The  partial  differential  equations  are,  then, 

T{D^  w,m)  =  n  (-  n^w,D^  w)  +  II, 

324.  When  the  independent  elements  of  position  are  the  rec- 
tangular coordinates  of  the  bodies,  these  equations  become,  by  the 
notation  of  (543i), 

^n.{l  nv)  =  2n{D^v,x)  +  211, 

:^,M^"  +!/"  +  ^")  =  2f2(- D,,W,  Id^.w)  +  2IL 


—  172  — 

S^mix''  +  S/"  +  n  =  ^^{t,^.^.'Q)  +  ^^.Q- 

325.  Through  the  preceding  investigations,  the  forms  are 
developed  by  which  every  dynamical  problem  can  be  expressed  in 
differential  equations.  It  only  remains,  therefore,  before  applying 
these  forms  to  especial  problems,  to  consider  those  methods  of  inte- 
gration which  are  best  adapted  to  their  discussion. 


CHAPTER    X. 

INTEGRATION  OF  THE  DIFFERENTIAL  EQUATIONS  OF  MOTION. 

326.  In  discussing  the  differential  equations  of  motion,  it 
might  be  permitted  to  suppose  a  previous  knowledge  of  all  that  has 
been  written  upon  the  integral  calculus.  But  since  the  profound 
philosophical  views,  with  which  this  subject  has  been  illuminated  by 
Jacobi,  have  not  yet  passed  from  the  original  memoirs  into  the  text- 
books, a  development  of  them  is  required  by  the  plan  of  the  present 
work  to  facilitate  its  further  progress. 

I. 

DETERMINANTS    AND    FUNCTIONAL    DETERMINANTS. 

327.  If  {n-{-iy  different  quantities  are  given,  which  are 
represented  by 


—  173  — 

in  which  every  number  from  0  to  n  can  be  substituted  for  k  or  for 
the  number  of  accents  denoted  by  i ;  and  if  all  possible  products  of 
{71  -}-  1)  factors  are  formed  similar  to 

in  each  of  which  the  same  number  is  never  repeated,  either  for  Jc  or 
for  i ;  and  if  these  products  are  successively  formed  by  mutually 
interchanging  two  of  the  inferior  numbers,  and  at  the  same  time 
reversing  the  sign  of  the  product ;  the  sum  of  the  products  has  been 
called  by  Gauss  the  determinant  of  the  given  quantities,  and  may  he  repre- 
sented hj 

^^  =  :E±ad^di a^:\ 

Thus,  for  example, 

•3^0  =  -^i  ^^=  ^? 

^1  =  ^  +  adx  =  aa\  —  a^d, 

9^2  ^^  -^  i  ^^1^2'  =  ad^a'l  —  ad<id[  -j-  a^d^d' 

—  aid a^  -\-  a^d ai  —  a^did'. 

The  same  result  might  also  have  been  produced  by  mutually 
interchanging  the  accents  without  disturbing  the  inferior  numbers. 

328.  The  sign  of  the  determinant  would  be  reversed,  by 
reversing  the  sign  of  the  product  originally  assumed  as  the  basis  of 
the  subsequent  changes. 

329.  If,  for  the  different  values  of  Jc,  all  the  given  quantities 
are  equal,  so  that 

af  =  a^, 

the  determinate  vanishes.  For,  by  interchanging  k  and  //  in  all  the 
terms,  the  sign  of  the  determinant  is  reversed  by  the  regular  process 
of  formation,  whereas  if  k  is  substituted  for  //  and  the  reverse,  no 


—  174  — 

change  is  produced  on  account  of  the  equality  of  the  given  terms. 
Hence 

or 

330.  Whenever  all  those  values  of  the  given  elements  vanish, 
for  which  i  is  as  great  as  7n,  while  k  is  less  than  m,  which  condition 
may  be  denoted  by  the  equation 

«<!>--!)  ==0, 

the  form  of  the  determinant  may  be  simplified.  For  it  is  evident 
from  inspection  of  the  fundamental  product, 

regarded  as  separated  into  two  factors,  that  every  elementary  prod- 
uct, produced  by  an  interchange  between  the  inferior  numbers,  such 
as  to  transfer  one  of  these  numbers  into  the  second  factor,  vanishes, 
and  may  be  neglected.     Hence 

^,=:E±aaw; «ir-i'^:^±«ir«tr4^i'^ a':' 

if 

331.     When,  in  the  preceding  proposition,  m  is  equal  to  n,  so 


that 

it  becomes 

for,  in  this  case, 


—  175  — 

332.  When,  in  addition  to  the  preceding  equation,  the  values 
of  the  elements  vanish,  for  which  m  is  equal  to  n  —  1,  so  that 

f,(i>n-\)  (-) 

"A<n  — 1     '-'j 

the  value  of  the  determinant  becomes 

333.  Whenever  the  equation  (174io)  is  true  for  all  values  of 
m,  it  may  be  written  in  the  form 

and  the  determinant  is  reduced  to  the  single  term 

<3l,„=.«a;4' a'"). 

334.  If  a  determinant  is  formed  from  the  given  elements,  with 
the  omission  of  all  those  of  w^hich  the  number  of  accents  is  i,  and 
those  of  which  the  inferior  number  is  k,  so  that  n  is  the  number  of 
factors  of  each  elementary  product,  this  determinant  is  the  factor  of 
«i:'  in  the  expression  of  the  determinant  ^„.  If,  therefore,  this  par- 
tial determinant  is  denoted  by  Q^f,  the  expression  of  the  complete 
determinant  is 


The  derivative  of  this  expression  is 
whence 


i^aCO^,=:Q/t('); 


335.     The  preceding  notation  gives 


=  "Slj^  „  =  ^  +  a[a2 


a 


(«) 


Hence  the  expression  of  qAl^'  can  be  deduced  from  that  of  Q^ 


—  176  — 
by  putting 

and  those  of  —  Q^**',  or  of  —  Q^*,  are  deduced  from  that  of  Q^  by 
putting 

^^  0,  or  k  =  0. 

336.  If,  in  the  third  member  of  (ITSgi),  «i/  is  substituted  for 
a^^\  the  expression  of  the  determinant  is  that  which  corresponds  to 
the  case  of  §  329.     Hence, 

and,  in  the  same  way, 

337.  If  a  partial  determinant  is  formed  from  the  elements  of 
Qk^^\  with  the  omission  of  those  in  which  the  number  of  accents  is  /, 
and  those  of  which  the  inferior  number  is  k',  this  determinant,  taken 
with  its  proper  sign,  is  the  factor  of  a^p  in  the  value  of  Q/t^'.  If, 
then,  it  is  denoted  by  Q^i!;^',  the  value  of  Q/t^!'  is 

in  which  it  must  be  observed  that,  from  the  definition 
These  equations  give 

338.  All  the  given  elements  which  have  k  or  k'  for  their  infe- 
rior number,  are  excluded  from  the  value  of  Q^^li'y,  and,  therefore, 


1 


t  t 


this  partial  determinant  is  not  affected  by  the  interchange  of  Ic  and 
//,  by  which  the  terms  of  the  complete  determinant,  comprehended 
in 

are  transformed  into  those  comprehended  in 

But  this  last  aggregate  of  terms  is  also  represented  by 
Hence  these  partial  determinants  satisfy  the  equations 

Qk^p  =  —  Q^i>:2  =  —  <^^^  =  ^1?;^ 

The  determinant  may,  therefore,  be  written  in  the  form 

339.  The  solution  of  linear  algebraic  equations  is  easily 
accomplished  by  the  aid  of  determinants.  For  if  the  given 
{ji-\-\)    equations   are 

u=^at-\-aJy-{- +  a  J,,  =  2:,{ajj,), 

it=at  -\-  a[t^  + +  cOn  =  ^k{(hh), 


u 


n)  ^  a^n^t  +  a^^t,  + +  «lr>4  =  ^..(«i"'4)  ; 


the  sum,  obtained  by  adding  the  products  of  the  given  equations, 
multiplied  respectively  by  q4^.,  o/t^., q4^">,  is 

^„4  =  ^''^^kU  +  ^W  + ^t'u'"'  =  ^ii^M'u'"') 

23 


—  178  — 

340.  If  all  the  quantities  ii,  u',  n",  etc.,  vanish,  i,  t^,  t2,  etc.,  must 
likewise  vanish,  unless  the  determinant  vanishes.^  If,  therefore, 
either  of  the  quantities  /,  /i,  ^25  etc.,  does  not  vanish,  when  it,  u',  ii', 
etc.,  vanish,  the  determinant  must  also  vanish,  whence  the  equation 
( 17613)  applies  even  when 

i'  =  /, 
or  for  all  values  of  z' 

Hence,  it  is  evident  that 

t-.t^-.t^ :  /„  =  oM) :  04'/) :  ok^ :  (M^. 

341.     The  process,  by  which  the  value  of  /„  was  obtained,  may 

be  regarded  as  designed  to  eliminate  the  n  quantities  t,ii,t2 

4_i  from  the  given  equations.     By  precisely  a  similar  process,  the 

m  quantities  t,  t^.  t^ tm-\  ^^^^  be  eliminated  from  the    first 

m  -|-  1  of  the  given  equations,  and  the  form  of  the  resulting  equa- 
tion must  be 

Bu  +  B'i{  + +  i?"")w<-'  =  CJ,,  +  C'^.+i^.+i  + -VCJn, 

in  which 

In  the  same  way,  if 

r^  =  :E±Gk'A\oA'i Q^lf), 

the  quantities  «''"  +  ^',  2<<'"  +  2', ?(<"'  may  be  eliminated  from  those 

of  the  equations  (1773o)  which  give  the  values  of  4o  ^2  +  1? 4? 


—  179  — 
and  tlie  form  of  the  resulting  equation  is 

Eu  +  E'u'  + +  E^-hi^-^  ^  FJ„,  +  i^,.+iC  +  i  + +  FJ,,, 


in  which 


But  the  two  equations,  obtained  by  these  processes,  must  be 
identical  in  the  ratios  of  their  coefficients.     Hence 

or 


OV.  m  iJV.  n'  m-ri,  It  '-><i  >i  '  »i  + 1  j  /i 

or  by  extending  the  series  of  ratios  to  all  values  of  m, 
But  it  is  easily  seen  that 

'n,  H  "^  n    ^'-/i  — 1, 


and 
whence 


^1'"  Qjk'^'"' 


>\.  =  ^^r,  =  a1Rl-K 


A  repetition  of  the  same  process,  in  a  different  order,  upon  the 
given  equations  gives 

Hence 


—  180  — 

342.  The  ratio  of  the  values  of  r„  and  ri^,^  may  be  prefixed  to 
the  series  of  ratios  of  (ITOig)  in  the  form 

The  series  of  ratios  gives,  then, 

^^  »  •    ^^m  '  «  •    ^7n  I  m  + 1,  h  J 

or 

This  investigation  is  derived  from  Jacobi. 

343.  The  variation  of  a  function  of  the  quantities  represented 
by  «i.''  is  expressed  by  the  formula 

If,  then,  the  values  of  the  quantities,  denoted  by  «*'',  are  such 
that 

and  if  the  corresponding  values  of  t,t^,  .....  .  /„  are  denoted  by 

^*^', /]", t\^\  the  expression  of  /^^'  assumes  the  form 

and  therefore 

=  (^^„+^,.,(q4(:)(z;/,)). 

344.  If  the  given  quantities  are  such  that 


—  181  — 

it  is  readily  perceived  that 

and 

whicli  is  given  by  Jacobi. 

345.  A  system  of  equations,  similar  to  those  of  §339,  repre- 
sented by  the  form 

gives,  in  the  same  way, 

If 

an  equation  similar  to  (ISOge)  is  derived, 

346.  Let   the    {n-\-lY   quantities,   represented    by   c^l\   be 
derived  from  the  given  elements  a^l^  and  Z»i!*  by  the  formula 

and  let  the  determinant  of  these  quantities  be 

-a^„=^±.e;4' c\:\ 

If  only  one  term  is  taken  in  each  of  the  quantities  ciJ',  the 
general  term  of  3„  is  represented  by 

A-fAm)   .(m').Amf')  ]Am)Um')Umf') 

^(li      ilj/      a  i//       U 1^     0  i^f     O  i^f      


—  182  — 

A  mutual  interchange  of  the  letters  A",  followed  by  a  mutual 
mterchange  of  the  letters  i  in  the  resulting  terms,  j^rocluces  all  the 
terms  of  ^„,  which  correspond  to  the  same  combination  {31)  of 
accents  m,  m^ ,  etc.  A  different  combination  of  accents  gives  a  dif- 
ferent set  of  terms  ;    and  if 

w^'"^  =^^±  «""'«f ''«r'"» «j::f  ^ 

q^(M)  ^  ^  4_  j(«)^(-.')^(«/o li^'p^ 

denote  the  determinants  of  the  given  elements  corresponding  to  one 
of  these  combinations,  the  complete  determinant  is  expressed  by 

=-^» ^  ill  V  '^a       ^  n     )} 

which  is  given  by  Jacobi. 
347.    In  the  case  of 

27  =  11, 

there  is  only  one  combination  (31)  of  the  accents,  so  that  in  this 
case 

n  n       n  7 

which  was  given  by  Cauchy. 
When 

there  is  no  combination  (31),  in  which  all  the  accents  are  different 
from  each  other,  and,  therefore,  it  follows  from  §  329  that,  in  this  case 


^ 


0, 


and  that,  in  all  cases,  the  combination  (31)  must  consist  of  accents 
which  differ  from  each  other. 

348.     In  the  special  case  of 


«i:'  =  ^i;', 


—  183  — 
which  gives 


cl!)=c|^\ 


the  value  of  the  determinant  is  reduced  to 


which,  'when 
is  reduced  to 


2^  =  n 


FUXCTIOXAL    DETERMINANTS. 


349.  If  the  given  elements  a^l^  are  the  derivatives  o^  (n -\-  1) 
functions  f,fi, /^  of  (;i  -]-  1)  variables  x,Xi, :r,j,  so  that 

the  determinant  of  the  elements  is  called  ihc  functional  deierminant  of 
the  given  functions.  Thus,  in  the  present  case,  all  the  terms  of  the 
determinant 

are  obtained  either  by  a  mutual  interchange  of  the  variables,  or  by 
a  mutual  interchange  of  the  functions,  the  interchange  being 
accompanied  in  either  case  with  a  reversal  of  the  sign,  precisely  as 
in  deducing  the  terms  of  the  ordinary  determinant.  The  proposi- 
tions, which  have  already  been  given  in  reference  to  determinants, 
are  easily  applied  to  functional  determinants. 

350.  In  the  case  in  which  all  the  functions,  above  the 
{in  -j-  l)st,  are  free  from  the  first  m  variables,  the  condition  of  (1749) 
is  satisfied,  so  that  the  notation  of  (17422)  gives  the  equation  (1742o) 

n  771  —  i  ?W ,  71  ' 


—  184  — 

351.  In  the  case  in  Avliich  every  function  is  free  from  the 
variables  of  which  the  inferior  number  is  less  than  that  of  the 
function  itself,  the  equation  (175io)  is  satisfied,  and  the  functional 
determinant,  reduced  to  a  single  term,  is 

%.  =  DJD^J,D^J,^ A„A. 

352.  If  the  given  functions  are  not  independent  of  each  other,  the 
determinant  vanishes.  For  if  the  equation,  which  denotes  their  mutual 
dependence,  is  expressed  by 

77=0, 

its  derivatives,  with  regard  to  the  given  variables,  are  represented 
by  the  equation 

The  equations,  included  in  this  form,  are  identical  with  the 
linear  equations  of  §  339  when  the  values  o^ii  vanish  and 

All  these  values  of  ^  cannot  vanish,  because  the  equation,  which 
expresses  the  mutual  dependence  of  the  functions,  must  involve  one 
or  more  of  them ;  and,  therefore,  the  determinant  must  vanish 
by  §  340. 

353.  If  either  of  the  given  functions  (f)  contains  any  of  the  other 
functions,  these  functions  mag  J)e  regarded  as  constant  in  finding  the 
fmictional  determinant.     For  each  derivative  of /^  is  the  sum  of  two 

parts,  one  of  which  is  derived  by  direct  differentiation  w^ith  refer- 
ence to  the  variable  explicitly  contained  in  the  function,  and  the 
other  part  is  obtained  by  indirect  differentiation  through  the 
functions  involved  in  /.  The  Avhole  determinant  may  then  be 
regarded  as  composed  of  two  such  portions.  But  the  portion  of  the 
determinant   obtained   by  the   indirect  differentiation  of/^  is  the 


—  185  — 

same  as  if/,,  not  containing  explicitly  any  variables,  were  .simply 
a  function  of  the  other  functions.  This  portion  must,  therefore, 
vanish,  and  the  remaining  portion  of  the  determinant  is  that  which 
is  obtained  by  direct  differentiation,  conducted  as  if  the  functions, 
involved  in  /,  were  constant. 

This  proposition  is  applicable  even  where  several  of  the  given 
functions  contain  the  remaining  functions;  but  not  when  they 
mutually  involve  each  other. 

354.  If  the  second  of  the  given  functions  contains  the  first, 
if  the  third  contains  the  first  and  second  functions,  and  if,  in  general, 
each  function  contains  all  the  previous  functions,  the  preceding 
proposition  is  applicable.  Hence  if,  b}^  means  of  the  first  function, 
the  first  variable  is  eliminated  from  all  the  other  functions ;  if,  by 
means  of  the  second  function  thus  reduced,  the  second  variable  is 
eliminated  from  all  the  subsequent  functions ;  and  if  this  process  is 
continued  until  each  function  is  liberated  from  all  the  variables 
designated  by  an  inferior  number,  although  it  may  involve  all  the 
preceding  functions  ;  the  determinant  is  reduced  to  a  single  term  as 
in  §  351.  This  will  often  afford  a  convenient  method  of  obtaining 
the  functional  determinant. 

355.  In  performing  the  successive  eliminations,  the  operation 
must  not  be  restricted  to  any  prescribed  order  of  the  variables,  but 
one  of  the  variables,  remaining  in/,  must  occupy  the  place  of  a:,. 
Hence  there  is  not  one  of  the  factors  of  the  determinant  in  the 
form  of  §  351  which  vanishes,  unless  a  function  be  obtained  from 
which  all  the  variables  are  explicitly  eliminated,  or,  in  other  words, 
unless  one  of  the  given  functions  is  included  in  the  others  and  can 
be  derived  from  them,  so  that  they  are  not  independent  of  each 
other.  If,  therefore,  the  given  functions  are  mnlmlly  independent,  their 
functional  determinant  does  not  vanish. 

356.  li  F,F^, F,,  are  given  functions  oi  ff, fy, 

24 


—  186  — 

which  are  themselves  functions  of  the  variables  a;,  Xi, x„,  the 

derivatives  of  the  functions  (i^)  with  respect  to  the  variables  (x;) 
are  represented  by  the  equation 

This  equation  coincides  with  (I8I24),  if  the  notation  for  «^'  is 
combined  with  the  notation 

The  remaining  notation  and  conclusions  of  §§346  and  347 
may,  therefore,  be  a^Dplied  to  this  case.  Hence,  by  (182i8)  ilie 
functional  determinant  of  the  independent  functions  {F^,  taken  tvith  respect 
to  the  same  number  of  varialiles  [x,),  loldcli  enter  into  {F^  only  as  they  are 
involved  in  the  same  number  of  independent  functions  (f)  explicitly  involved 
in  (Fi),  is  obtained  by  nmltiplying  the  functional  determinant  of  (i^)  taken 
tvith  respect  to  (f)  by  the  functional  determinant  of  (f)  taken  tvith  respect 

to  {Xi). 

If  the  number  (p  -\-l)  of  functions  (f)  exceeds  the  number  {n -\-  1) 
of  functions  {Fj),  the  complete  functional  determinant  of  [Fj)  is  by  (182n) 
the  sum  of  all  the  partial  determinants  cf  (Ff)  obtained  by  every  possible 
combination  of  {n-\-\)  of  the  functions  (f). 

If  the  number  of  functions  {f)  is  less  than  that  of  the  functions  (Fi), 
the  fimctional  determinant  vanishes,  as  in  (I8225),  tvhich  corresponds  to  the 
proposition  that  the  number  of  independent  functions  cannot  exceed  the  num- 
ber of  variables,  by  tvhich  they  may  be  expressed. 

357.     In  the  case,  in  which 

F  —  X- 
all  the  derivatives  of  {F^)  with  reference  to  the  variables  {Xi)  vanish, 


—  187  — 
except  those  included  in  the  form 

In  this  case,  therefore, 

is  the  functional  determinant  of  (a-,)  regarded  as  functions  of  (/,), 
and  the  equation  (182i8)  becomes 

or  the  functional  determinant  of  (rr,)  taken  vntli  respect  to  (/,)  is  the  recipro- 
cal of  the  functional  determinant  of  [fi)  taken  toith  respect  to  [xi). 

358.  If  in  the  linear  equations  of  §  339,  the  values  of  {t)  are 
expressed  by  the  formula 

either  of  the  equations  is  represented  by 

unless 

m  =  i, 
in  which  case 

This  value  substituted  in  (ITTs))  gives 

359.  If  it  is  again  assumed  that 


188  — 

the  equations  of  §  345  give 

^,/<f-)  =  2,Df,df,  =  ^  =  ^  log  ^0,^. 


'^ 


360.  By  the  same  process,  it  may  be  proved  that,  if  (/,)  are 
the  variables  and  (Xj)  the  independent  functions, 

^,i)./.^  =  d  log  ^„  =  —  (Tlog  ^o„. 

But  it  must  be  observed,  that  in  finding  the  derivatives  of 
dxk  they  are  supposed  to  be  expressed  as  functions  of  the  original 
variables,  precisely  as  in  the  preceding  section  the  values  of  d  fj^ 
are  supposed  to  be  expressed  in  terms  of /t. 

301.     The  equation  (I889)  reduced  to  the  form 

may  be  added  to  the  identical  equation 

The  sum  is,  by  (ISTs;), 

^  =  2:,D.,{:W.Jx:) 

362.  In  the  case,  in  which  the  arbitrary  variation  d  is  assumed 
such  that 

except  for  the  value 

/=0, 


—  ISO  — 
the  preceding  equation  becomes 

If  this  equation  is  multipHed  by  /  and  added  to  the  equation 

the  sum  is 

3G3.  If  the  equations,  by  which  the  functions  (/)  depend 
upon  the  variables  (2',),  are  represented  by 

their  derivatives  are  represented  by 

Tlie  comparison  of  this  equation  with  (1864)  indicates  that  tlio 
concluding  propositions  of  §  356  may  be  applied  to  this  case,  pro- 
vided the  negative  sign  is  introduced  as  a  factor  of  all  the  deriva- 
tives taken  with  respect  to  (/,).  Hence,  if  the  number  of  the 
functions  (/,)  is  the  same  with  that  of  the  variables  (.r,), 

^  +  D^FD.^F, D^^F,,  =  (— )"+i  3[.„^  +  D,FDf,F^ ^/„i^„, 

and 

^' «  —  I      J        '2:+  Dj  FJ)f,  F, Z»/„  F,  ' 

364.  If  the  number  of  the  functions  (/)  exceeds  that  of  the 
variables  (.r,)  and  is  ^;  +  1  instead  of  «  -|-  1,  let  {F})  be  the  form  of 
(i^)  when  the  last  p  —  u  of  the  functions  (/)  are  eliminated  from 
it  by  means  of  the  last  p  —  n  of  the  given  equations.     In  this  case 


—  190  — 

it  follows  from  the  reasoning  of  §  354  that 

v+7>/^Z>^/;  D.r  F  Df     ]?       Df  F 

=  -  +  D^F'l).r,Fl Dx^Fl  ^  ±  ^/ ,:i^.  +i^/„..i^.+2 ^f,K. 

2±DyFJ)f^F, ^/^„ 

_  -^  4-  n  Fi Df   F]  Df  F^^A-  Df     F  ^^Df     F  ^o Df  F 

But  the  equation  (ISOge)  is  applicable  to  this  case  if  (i^)  is 
changed  to  {F}),  and,  therefore,  the  introduction  of  a  common  factor 
into  the  terms  of  (1892g)  gives,  by  means  of  the  preceding  equations, 

6S    _  (      y,  +  1  ^±D^FD^,  F, D^,,F^  Df„+,  F„^, 

»— V      >*  ^±D,FD/iF, Dr^F^, 


o 


65.     There   are  various  interestino;  and  instructive  relations 


o 


between  the  partial  determinants  of  functions  which  have  been 
developed  by  Jacobi,  and  which  will  be  found  useful  in  discussing  the 
theory  of  differential  equations.  If  the  number  of  the  functions 
(/j)  as  well  as  of  the  variables  (.r,)  is  increased  to  ni  -\-  n  -|-  1 ,  let 

mi)  —  ^4-7)  fDx  f,  Dc       f     .Dx       /  ,  . 

If,  then,  from  the  function  (Z,,^,),  all  the  variables  x,  x-^, 

rr„_i  are  eliminated,  and  the  functions  /, /i fn-i  introduced 

in  their  places,  and  the  function  {fn+i)  thus  transformed  is  denoted 
by  {f\j^i),  the  values  of  0^  become 

2B'')  =  gic     ,Dx      /-i... 

k  n  —  1         n->rkJ  n-\-i  • 

The    determinant    of   the    (m  -{-If  functions   (^i^')    is,   con- 
sequently^, 

^  +  ®5B;2B;' S^c")— .'7|5'«+i  v-i-z>;r  f^Dx     /•!    -,      .  D.r      /I      . 

-!-  1       ^  in    JVun  — 1  —  _L         nJ  «         n  +  lJ  «  +  l «  +  m./  n-\-m' 


—  191  — 
But  it  is  obvious  that 


whence 


^  +  ^% ^\:p  =  vji:r-i  ^'. 


+  m ' 


366.  If  '^'k^  denotes  the  value  which  3^'^„_i  assumes  when  all 
the  derivatives  relatively  to  Xi  are  changed  into  the  derivatives  rela- 
tively to  x,,^;,,  it  is  evidently  the  factor  of  ^-e^fn^i  in  the  value  of 
—  ^J;!'.     In  the  value,  therefore,  of  the  determinant 


^  +  2B'5'); SB'-), 

the  factor  of  D,f„^-rJ],  +  i ^■'■nfn  +  m  is 

^y-+-':s±%%[ ^^1^^ 

But  the  factor  of  the  same  quantity  in  ^^i+m  is?  by  inspection, 
( \m+i-^\j)^  fD^      f  Br       f  Dx       /•    ,  , Dx      f     , 

\  )  ^    _L  nJ  n  +  lJ  I n  +  mJm         ,„  +  i./w!  +  l «     i  J  n —\ 

It,  therefore,  follows  from  (lOl^)  that 

^  +  SB  S^i '5]',;;" 

\  /  w  +  l'^    J_  m  +  iy  m  +  'lJ  I m+nJn  —  \' 

367.     The  factor  of  ^.^._J„  +  ,  in  the  value  of  5]';' is —'(!<; -^S 
and  therefore  the  determinant 


—  192  — 

is  in  (I9I12)  the  factor  of  D,J,,^^  ^\fn-\-i ^\.-Jn^m-    But  the 

factor  of  this  same  quantity  in  '3^"„4-„;  is,  by  inspection, 

\         /     ■""   _L  n  +  lJ  «  +  lV  1 n  +  »i  J  m  —  l         „J,n n  J  n 

V  /  — i-  mj  m  +  lJ  I m  +  nJ  n' 

Hence  it  follows  from  (lOl,,)  that 

^  +  33^^,qi; f€""-i) 

'       J.  ^  •  •  •  •  •  •    ),i 

=  (-)"+^  -m::^-!^  ±  ^^„./^-^„,,/\ ^,„,„/;. 

368.     By  the  same  j^rocess  it  will  be  found  that,  in  general, 

:r  +  '33  0];  ^:; g](.'"-i)^f /c' , ,         ^c^'-d 

309.     If  the  flxctor  of  93^.  in  the  value  of  (I9I29)  is  denoted  by 
( — y^-kj  this  expression  gives 

^  ±  ^%% ^ti;r-^^  =  ^,(^,213,), 

in  which  neither  the  quantities  ('C(;'),  nor  any  function  of  them,  such 
as  K,  contain  the  derivatives  of/,,.  Hence  the  derivative  of /„,  with 
respect  to  .r„+i.,  only  occurs  in  this  expression  because  it  is  in  SB^.,  in 
which  its  coefficient  is  ^^,_i,  so  that  the  term  of  the  preceding 
expression  which  contains  this  derivative  is  L 'Sli  .D^  f  \?  ,,, 
is  the  coefficient  of  the  same  derivative  in 

^  +  i>.  /i).       / B.       /, 

mJ  m  + 1 ./  1  •  111 +n  J  It} 

the  equation  (1928)  gives 

H  [ )    ^     ^^n-lfljc' 


—  193  — 
The  comparison  of  (192y)  with  this  equation  gives 

0 

It  is  to  be  observed  that,  from  their  definitions,  the  functions 
fi^  and    2B^.  are    both    of  them   partial    determinants   of  the   same 

functions  /, /i, /„_i  the   former   being   taken  with  respect   to 

the  variables  a;,^,  ^m  +  i  ^n  +  m  excluding  n'„  +  i.,  and  the  latter 

being  taken  with  respect  to  the  variables  .r,  .Tj  . . . .  .r„_i  and  a^^+k- 

In  the  case,  therefore,  in  which  m  and  n  are  equal,  these 
two  determinants  are  formed  with  respect  to  an  entirely  different 
set  of  variables,  and  each  of  the  variables  x,,^^  is  taken  in  succession 

from  the  set  x„,  x„+i 2:,^  in  forming  Uf,  and  combined  with  the 

set  ^,  .Ti ^n-i,  in  forming  2B^.. 

370.  The  first  member  of  (lOSg)  does  not  contain  any  deriva- 
tive of /„  with  respect  to  a  variable  of  which  the  inferior  number 
is  less  than  m.  The  factor,  therefore,  of  such  a  derivative  as  Z>^/„ 
in  the  second  member  vanishes  identically  ;  whicli  is  represented 
by  the  equation 


V 


371.     If  in  the  equation  (I9I3) 

this  equation  becomes,  by  writing  n  —  1  for  m, 

But 

25 


—  194  — 

so  that  if  X  is  supposed  to  l^e  a  function  of  the  other  variables  and 
/  to  be  equal  to  x,  these  equations  are  reduced  to 

^,  (Q/b,ft-./)  =  2,(c^h°'.^)  =  ^  +  ^-,/I^' J5 D'.fl 

1 

in  which 

Qk  =  ^,^,  ,, 

and,  by  (ITGg),  — ■'^\  is  deduced  from  Q^  by  changing  the  deriva- 
tives relating  to  .r^.  into  the  derivatives  relatively  to  x.  This 
equation  is  derived  from  Lagrange. 

372.  In  the  greater  portion  of  these  formuloB  upon  functional 
determinants,  the  derivative  taken  Avith  regard  to  either  of  the 
variables  may  be  supposed  to  be  frequently  repeated,  so  that  Z^^ 
may  be  substituted  for  D^  ,  and  h  may  even  be  zero.  Thus  if,  in 
§  365,  D\  is  substituted  for  D^,  and  if 

«  =  1, 
the  equations  of  that  section  are  reduced  to 


Hence  if 


and  \in  is  written  for  wi  +  1,  the  equation  (lOOgi)  becomes 


—  195  — 

If  each  of  the  functions  (/,)  is  raultipHecl  by  /,  the  values  of 
the  functions  (e',)  remain  unchanged,  and  therefore  the  value  of  the 
determinant 

^±fD.^nD.,f, D.,J^^ 

is  multiplied  by  t''^^. 

373.  A  system  of  functions  (y^)  can  always  be  found  such  that 
their  determinant,  with  respect  to  the  variables  (.r^),  may  be  equal 
to  a  given  function  IT  of  those  variables.  For,  if  all  these  functions 
except  f,^  are  assumed  at  pleasure,  and  if  f\  represents  the  form 
of  f„  when  all  the  variables  except  x,^  are  eliminated  and  the 
remaining  functions  (/j)  are  introduced  in  their  place,  the  required 
determinant  becomes 

Hence, /}j  is  by  (187io)  determined  by  the  integration 

in  which  it  must  be  observed  that  the  quantity  under  the  sign  of 

integration  is  expressed  in  terms  of /,/i fn-\  ^^^  ^n- 

In  the  case  of 

n=i 

this  formula  becomes 


f\  =1  s::,  =X  3^.-.. 


The  substance  of  all  these  investigations  upon  determinants  is 
taken  without  important  modifications  from  Jacobi. 


—  196 


MULTIPLE    DERIVATIVES    AND    INTEGRALS. 

374.  The  functional  determinant  is  shown  by  Jacobi  to  be  of 
sinouhir  use  in  the  transformation  of  multiple  derivatives  and  inte- 
grals. The  expression  of  these  functions  is  facilitated  by  the 
notation 

r\  n-m  +  l  7^"  -»'  +  !  / 

■^fm  ■■■■     -^fmJm  +  l Jn, 


and 


If  then 


n  =  i>;+i  TF, 


a  new  variable  x„,  which  is  a  given  function  of  all  the  variables,  /^ 
may  be  substituted  for  either  of  them  as  /„  in  W,  and  the  new 
derivative  is  given  by  the  formula 

Another  new  variable  :?;„_i  may  next  be  introduced  instead  of 
fn-\  in  the  same  way,  and  this  process  may  be  repeated  of  substi- 
tuting successively  for  each  variable  fi  a  new  variable  x^,  which 
shall  be  a  function  of  all  the  other  variables  remaining  in  the 
derivative  at  the  instant  of  the  substitution  of  oCi,  until,  finally,  an 
entirely  new  set  of  variables  shall  be  introduced  into  the  derivative. 
The  final  form  is 

^'-DJD.j^ D^j^  =  Dl^^W. 

From  the  comparison  of  this  form  with  §  351,  it  appears  that 


—  197  — 

the  factor  of  £1  is  identical  with  the  determinant  of  that  section. 
From  the  reasoning  of  §§  353  and  354,  it  follows  that  the  determi- 
nant is  not  changed  by  substituting  in  either  of  the  quantities  (/) 
regarded  as  functions  of  the  variables  (.r,)  the  values  of  any  or  all 
the  preceding  functions  in  terms  of  these  variables.  But  each  of 
the  functions  (/,)  contains,  in  its  present  form,  none  of  the  succeed- 
ing functions ;  so  that,  after  this  substitution,  it  is  expressed  in  terms 
of  (.r,) .     Hence 

375.     The  preceding  equation  gives,  for  the  multiple  integral 

'rt  +  l  /•"  +  ! 


in  which  the  limiting  values  of  [xi)  may  be  supposed  to  be  constant, 
while  those  of  (/)  may  not  be  constant.  If  then  77  is  determined 
by  the  integration 

so  as  to  contain  neither  of  the  variables  (.r,)  except  as  they  are 
involved  in  (/),  it  is  by  §  353  unnecessary  to  have  regard  to  the 
derivatives  of  77  otherwise  than  as  they  are  dependent  upon  /  in 
finding  the  value  of  the  determinant,  wdiich  is  the  first  member  of 
the  following  equation,  and  Avhich  therefore  becomes 

^±D^nD.  J.D.J, D.j^  =.  ^..Djn=  %^S2. 

Butby  (ISOs) 


—  198  — 
and,  therefore, 

in  which  /w«.y  denotes  that  the  function  to  which  it  is  prefixed  is 
referred  to  the  Hmiting  values  of  x,,,  so  that  the  difference  of  the 
values  of  the  function  at  these  two  limits  is  represented  by  this 
notation. 

But  since 

it  is  evident  from  (lOTuj)  that 


JJ    lim.,(7r^t)  =  lim./^"     n- 


and  a  similar  equation  may  be  given  for  each  of  the  terms  of  the 
last  member  of  (1 983),  whereby  this  equation  is  reduced  to 


J^  /!  + 1  pn 


The  multiple  integral  of  the  (;z  -|-  1)  th  order  is  thus  reduced  to 
2ii-\-2  multiple  integrals  of  the  nth.  order,  and  this  reduction  may 
be  continued  until  the  whole  process  is  made  to  depend  upon  single 
integrals,  of  which  one  is  performed  with  reference  to  /,  and  the 
number,  performed  with  reference  to  any  other  of  the  variables  (/j), 
is 

2'(«  +  l);« (^ii-\.2  —  i). 


—  199  — 


II. 

SIMULTANEOUS    DIFFERENTIAL    EQUATIONS    AND    LINEAR   PARTIAL   DIFFERENTIAL 
EQUATIONS    OF    THE    FIRST    ORDER. 

376.  An  ef|uation 

/=o, 

of  which  the  derivative  vanishes  identically,  by  means  of  the 
simultaneous  differential  equations  represented  by 

Dxi  —  Xi, 

in  which  (X,)  are  given  functions  of  the  variables  [xi),  is  called  an 
iniegral  of  these  equations.  It  is  a  general  integral  if  it  involves  arbi- 
trary constants,  and  a  imriknlar  integral  if  it  does  not  involve  arbi- 
trary constants.  When  it  involves  an  arbitrary  constant,  it  is  more 
conveniently  expressed  in  the  form 

f—u, 

in  which  a  is  an  arbitrary  constant. 

377.  A  function  /,  which  satisfies  the  linear  partial  differential 
equation  of  the  first  order 

is  called  a  solution  of  this  equation.     By  means  of  the  notation 

0 


—  200  — 
this  equation  may  be  written 

378.  The  first  member  of  ever?/  integral,  expressed  in  the  form  (lOOg) 
or  (lOOig),  of  the  simultaneous  differential  equations  (lOOn)  is  a  solution  of 
the  fartial  differential  equation  (2OO2)  ;  and,  conversely,  every  solution  of 
the  partial  differential  equation  (2OO2)  is  the  first  member  of  an  integral  of 
the  simultaneous  differential  equations  (IDQg),  and  its  second  member  is  any 
constant.  For  the  derivative  of  (lOOg)  or  (lOOig)  vanishes  by  the 
substitution  of  (lOOn),  which  gives 

that  is,  /  satisfies  the  equation  (2OO2).  Reciprocall}^,  the  satisfying 
of  this  condition  is  all  that  is  required  in  order  that  (lOOji,)  may 
be  an  integral  of  (lOOg). 

379.  If  the  equation  (199g)  is  solved  relatively  to  x,  so  as  to 
express  x  as  a  function  of  the  other  variables  (rr,),  the  equation 
( 19925)  becomes 

X— Fi,„.T  =  0, 

which  is  distinguished  from  (2OO2),  because  the  function  x,  of  which 
the  derivatives  are  taken,  is  involved  in  the  functions  {Xi),  whereas 
/is  not  involved  in  these  functions. 

380.  A  solution  of  (2OO2)  which  shall,  for  a  given  equation 
between  the  variables,  become  equal  to  a  given  function,  may  be 
determined  by  means  of  series.  For  this  purpose,  let  the  given 
equation  be 

(^  =  T, 

in  which  t  is  constant,  and  t  a  function  of  the  variables,  and  let  the 


—  201  — 

solution  become  a  function  (p  of  the  variables  when  this  equation  is 
satisfied.  If  then  t  were  assumed  to  be  also  one  of  the  variables 
of  the  given  equation,  and  such  that  in  forming  the  simultaneous 
equations 

I)t=l, 

by  which  the  simultaneous  equations  become 
and  the  given  partial  differential  equation  is 
assume  the  functional  notation 

n  =  —  f/n, 

and  the  integral  of  the  partial  differential  equation  with  reference 
to  i  is 

/— 9  — □/=o. 

which  gives 

f=T^  =  (1  +  D  +  n=  +  etc.)'; 

0 

This  value  of  /  taken  from  Cauchy,  expresses  a  true  solution  of 
the  given  equation  if  \Z\'(p  is  finite  for  all  values  of  i  and  vanishes 
when  i  is  infinite,  which  is  always  the  case  for  sufficiently  small 
values  of  i  —  t  . 

381.  There  are  n  independent  soluiions  of  the  partial  differential 
equation  (2OO2)  and  no  more  than  n  indejjendent  solutions. 

26 


—  202  — 

First.  The  equation  (2OO2)  lias  n  independent  solutions.  It 
lias  been  proved  in  the  preceding  section  that  it  has  one  such  solu- 
tion.    Let  it  then  be  assumed  that  m  such  independent  solutions 

have  been   obtained,  denoted   by  /„,  /„_i /„_,„+i.      These 

independent    solutions    may   be    substituted   for   the   m   variables, 

X,,,  x„_x ^n-m+i^  with  regard  to  which  they  are  independent ; 

and  if /a/  denotes  the  value  of  /  when  expressed  in  terms  of  the 
new  variables,  the  equations  of  substitution  are  represented  by 

in 
n  —  »;j  + 1 

But  since 
the  substitution  of  these  equations  in  (2OO2)  reduces  it  to 

in  Avhicli  the  functions  /t  may  be  regarded  as  constant.  This 
reduced  equation  has,  then,  a  solution  by  the  preceding  section ; 

its  solution  does  not  involve  the  variables  ^„,  ^,i_i .r,i_,„-|-i, 

and  is  independent  of  the  given  m  solutions.  The  given  equation  is 
then  proved  to  have  another  solution  independent  of  the  given 
solutions ;   and  this  number  may  again  be  increased  by  the  same 

process,  until  the  n  independent  solutions  are  obtained,  /i,/2 /„. 

Secondly.  The  equation  (2OO2)  cannot  have  more  than  n  inde- 
pendent solutions.  For  if  there  are  {ii-\-Y)  solutions  (/,),  each 
gives  an  equation  represented  by 

which  may  be  regarded  as  a  linear  equation  between  the  quantities 


—  203  — 

(X;).  By  the  usual  process  of  elimination,  if  7)1,^  denotes  the 
functional  determinant  of  (/)  with  respect  to  the  variables  (x^), 
these  equations  give,  by  §  340, 

But  all  the  quantities  X^  do  not  vanish,  and,  therefore, 

«,  =  o, 

or  the  (n  -\-  1)  functions  (/,)  are,  by  §  355,  not  independent  of  each 
other. 

382.  It  is  evident,  from  the  preceding  demonstration,  t/iat  any 
function  of  the  solutions  of  the  linear  iiartial  diffe^'erd/ial  equation  (2OO2)  *'^ 
itself  a  solution  of  that  equation. 

383.  A  system  of  finite  equations,  of  which  the  derivatives 
are  satisfied  by  the  simultaneous  equations  (lOOis),  i;^  called  a  system 
of  integral  equations  of  the  simultaneous  differential  equations.  This  system 
is  said  to  be  genercd,  when,  by  the  successive  elimination  of  the  con- 
stants, it  can  be  reduced  to  a  form,  in  which  each  equation  involves 
an  arbitrary  constant  not  included  in  the  other  equations,  and  it  is 
complete  when  the  number  of  finite  equations  is  equal  to  that  of  the 
given  difierential  equations.  When  reduced  in  the  method  just 
proposed,  the  general  system  is  represented  by 

in  which  the  functions  (f/,)  are  independent  of  the  arbitrary  con- 
stants {[^i).  The  ixirticular  system  is  represented  by  a  set  of  similar 
equations,  combined  with  other  equations,  which  involve  no  arbitrary 
constants,  and  which  are  represented  by 

i^^^O. 


—  204  — 

384.  Each  equation  of  a  general  sf/stcm  of  mtcfjral  equations, 
reduced  to  the  form  (20824),  is  an  integral  of  the  given  sinmltaneoiis  differ- 
ential equations.  For  the  derivative  of  (20824),  when  reduced  to  a 
finite  equation  by  the  substitution  of  the  given  differential  equations, 
is  independent  of  the  arbitrary  constants  (fij),  and  vanishes,  there- 
fore, independently  of  the  equations  themselves  in  which  these  con- 
stants are  involved.  When  the  system  is  general,  therefore,  the 
functions  (g),)  are  functions  of  the  solutions  (/^)  of  the  partial  differ- 
ential equation  (2OO2). 

385.  If  the  system  is  particular,  and  if  the  number  of  the 
equations  (2O831),  which  are  free  from  arbitrary  constants  is  m  —  n, 
the  same  number  of  variables  can  be  eliminated,  by  their  aid,  from 
the  functions  (A")  and  (9^,).  The  equations,  to  which  (203ai)  are 
thus  reduced,  are  integrals  of  the  simultaneous  differential  equations, 
represented  by 

Dxi  —  Xi, 

in  which  the  variables  (x,),  of  which  the  number  is  m,  are  those 
which  are  not  eliminated  from  (X,)  and  (9),). 

386.  The  system  of  equations  (2O831)  is,  by  itself,  a  particular 
system  of  integral  equations  of  the  given  differential  equations, 
which  does  not  contain  any  arbitrary  constant.  For  the  derivative  of 
either  of  them,  involving  no  arbitrary  constant,  must  be  satisfied  by 
means  of  the  equations  (lOO^s)  and  (20831),  without  any  aid  from  the 
equations  (2O824).  The  derivative  of  each  of  the  equations  (20834) 
is,  for  the  same  reason,  satisfied  by  the  same  equations  (199i2)  and 
(2O831),  without  tlie  assistance  of  the  equations  (2O824). 

387.  The  functions  (/)  maj^  be  supposed  to  be  introduced  as 
the  variables  instead  of  the  given  variables  (.r^).  By  this  substitu- 
tion, tlie  proposed  system  of  difterential  equations  assumes  the  form 


—  205  — 

By  this  same  substitution  in  the  equations  (20824)  ^"^iid  (2033i),  the 
equations  (20831)  may  be  readily  reduced  by  processes  of  elimination 
to  an  equal  number  of  equations  of  the  form 

in  which  the  functions  (i^)  do  not  involve  those  of  the  functions  (/,) 
of  which  the  values  constitute  the  first  members  of  these  equations. 
Hence  the  derivatives  of  these  equations,  reduced  to  a  finite  form 
by  the  substitution  of  (2002)  become  of  the  form 

or 


x-^  I 


But  this  equation  does  not  involve  either  of  the  functions  (/,) 
which  are  not  contained  in  (i^),  and,  therefore,  cannot  depend  upon 
the  equations  (2058).  It  is,  therefore,  identical,  the  functions 
{Fi)  are  independent  of  .r,  and  the  equations  (2O831)  from  which  they 
are  derived,  contain  only  the  functions  (/j).  The  substitution  in 
(2O831)  of  the  arbitrary  constants  («,)  for  the  functions  (/)  to  which 
they  are  equivalent,  reduces  these  equations  to  conditional  equa- 
tions between  the  arbitrary  constants.  These  equations  (2O831),  there- 
fore, represoit  the  conditional  equations,  to  luhich  the  arl)itrary  constants  of 
the  integrals  o/"  (199i2)  nnist  he  sulject,  in  order  that  they  may  coincide  with 
the  particular  system  of  integral  equations,  to  ivhich  the  equations  (2O831) 
belong.  After  the  introduction  of  the  functions  (/),  instead  of  the 
variables  {x^),  into  the  functions  ((/,),  these  functions  ((f,)  can,  by  the 


—  206  — 

substitution  of  (2058)?  be  freed  from  all  the  functions  (/)  whicli  are 
not  contained  in  (i^).  The  derivatives  of  ((/),)  when  thus  reduced 
become,  by  means  of  the  equations  (2002),  of  the  form 

Dcp,  =  XD^(p,  =  0, 

which  must    vanish    independently  of   the   equations   (2058),  ^'^^^y 
therefore,  the  functions  {(pi)  do  not  involve  .r.    Hence,  hy  the  siihstitidion 
of  {x^  for  (/i)  the  equations  (20824)  9^'^^  i^^^  values  of  {fi^  in  terms  of  {0,^). 
088.     From  any  one  given  integral  equation,  denoted  by 

z«  =  0, 

the  whole  system  of  integral  equations,  to  which  it  belongs,  can  be 
readily  obtained.  For  the  finite  equation,  to  which  the  derivative 
of  this  equation  is  reduced  by  the  substitution  of  the  given  differen- 
tial equations,  is,  from  the  very  nature  of  the  problem,  another  of 
the  required  system  of  integral  equations.  The  derivative  of  this 
new  equation  gives  a  third  integral  equation,  and  the  continuation 
of  this  process  leads  to  the  final  ddcrmmaiion  of  the  ivholc  of  the  required 
system  of  integral  equations. 

389.  This  process  of  deriving  a  system  of  integral  equations 
from  one  of  its  component  equations,  affords  the  means  of  testing  a 
proposed  equation,  and  ascertaining  whether  it  be  an  integral  equa- 
tion. For  as  great  a  number  of  independent  integral  equations  is 
not  admissible  as  that  of  the  variables  themselves ;  if,  therefore,  the 
application  of  the  process  to  a  proposed  equation  conducts  to  a  numher  of 
indcjyeiideni  equations  equal  to  tJiat  of  the  variaUes,  it  is  a  sufficient  proof 
that  the  p>roposed  equation  is  not  an  integral  equation. 

390.  When  a  system  of  integral  equations  contains  superfluous 
arbitrary  constants,  that  is,  constants,  wdiicli  remain  in  the  functions 
(^j),  after  the  system  is  reduced  to  the  form  given  in  §  383  ;   such 


—  207  — 

constants  supply  the  means  of  obtaining  other  integral  equations 
which  are  not  contained  in  the  given  system.  Thus  if  (206n)  denotes 
an  integral  equation,  from  which  the  proposed  system  may  be  sup- 
posed to  be  derived,  so  that,  reciprocally,  this  equation  may  be 
derived  from  the  proposed  system,  and,  therefore, 

in  which  F  is  any  arbitrary  function  ;  and  if  the  notation  is  adopted 

in  which  arbitrary  constants  are  denoted  by  (;',) ;  the  equation 

is  also  an  integral  equation.     For  the  equation 

l)u  =  0 
gives,  by  direct  differentiation, 

nr^^u  —  o. 

But  it  is  obvious,  from  the  form  of  (2076),  Hi^t  the  derivatives 
of  u  with  reference  to  those  of  the  constants  (f:?,),  which  are  elimi- 
nated from  the  functions  (y,)  and  to  which  these  functions  are  equal, 
are,  themselves,  functions  of  ((/),•  —  /i?,)  and  i/'i ;  whereas  the  deriva- 
tives of  u  with  reference  to  the  superfluous  independent  constants 
(/?,),  which  are  contained  in  the  functions  (9,),  are  not  merely  func- 
tions of  ((pi  —  /:?,)  and  i/^^.  Hence  the  integral  equation  (207i3)  is  a 
new  equation,  if  it  contains  the  derivative  of  w  with  reference  to 
either  of  the  superfluous  constants  (fi,),  and  there  are  as  many  of 
these  new  equations  as  there  are  superfluous  constants.  But  the 
number  of  independent  integral  equations  thus  obtained,  is,  of  course. 


—  208  — 

subject  to  the  condition,  that  it  cannot  exceed  the  number  (n)  of 
the  independent  solutions  of  the  equation  (2OO2). 

391.  Of  all  systems  of  integral  equations,  that,  in  which  the 
arbitrary  constants  are  the  values  which  the  variables  themselves 
assume  for  a  given  value  of  one  of  them,  deserves  especial  consider- 
ation. To  simplify  the  discussion  of  this  case,  and  place  it  in  the 
position,  in  which  it  will  best  illustrate  the  problems  of  mechanics, 
the  variable  (.r),  of  which  the  value  is  given,  may  denote  the  ^inie, 
and  the  given  time  is  the  epoch  or  origin,  at  wdiich  the  elements  of 
the  system  of  variables  are  given,  and  from  which  the  variations  are 
estimated.  The  values  of  the  variables  at  this  beginning  of  time 
may  be  termed  tlieir  iuifml  values,  while  those  at  any  subsequent 
time  are  their  Jiual  values.  The  differential  equations  express  the 
laws  of  change,  under  which  the  variables  pass  from  their  initial  to 
their  final  values,  and  are  equally  compatible  W'ith  any  proposed 
combination  of  initial  values.  The  initial  rallies  are,  therefore,  ivholly 
arhitrary  and  independent.  Their  number  is  equnl  to  that  of  the  variables 
[xi),  and,  consequently,  equal  to  the  tvhole  number  of  independent  arbitrary 
constants,  luhieh  is  required  for  the  complete  integral  equations. 

The  epoch  is  also  arbitrary,  and  seems  to  introduce  an  addi- 
tional arbitrary  constant.  But  this  constant  is  obviously  superflu- 
ous ;  it  corresponds  to  the  arbitrary  position  of  the  problem  in 
time,  without  involving  any  modification  of  the  essential  conditions ; 
and  is  the  complement  of  the  arbitrary  element,  wdiich  is  not 
expressed,  and  in  reference  to  which  the  derivatives  in  the  equations 
(IQOia)  are  supposed  to  be  taken. 

392.  The  passage,  down  the  stream  of  time,  from  the  initial  to 
the  final  values,  conformably  to  the  conditions  of  change  expressed 
in  the  differential  equations,  may  be  imagined  to  be  reversed  and, 
in  a  retrograde  transit,  the  same  laws  of  change  would,  by  their 
reverted  action,  restore  the  variables  to  their  initial  values.     In  the 


—  209  — 

direct  action,  the  initial  values  constitute  the  cause,  and  the  final 
values  are  the  effect ;  whereas,  in  the  reverted  action,  the  final 
values  become  the  cause  of  which  the  initial  values  are  the  efibct. 
Hence  it  follows  that,  in  any  integral  equation  hetiveen  the  final  and  the 
initial  values  of  the  vanahles,  the  final  and  initial  values  of  ejich  vanahle 
may  he  mutually  interchanged,  and  the  resulting  equation,  if  not  identical  iiith 
the  given  equation,  is  a  new  integral  equation.  In  making  this  change,  the 
sign  of  the  variable,  tvhich  expresses  the  interval  of  time,  must  he  reversed, 
because  the  interval,  which  is  positive  with  reference  to  the  initial 
epoch,  is  negative  with  reference  to  the  final  epoch.  If,  indeed,  the 
interval  were  expressed,  bj  means  of  the  initial  value  [x^)  and  the 
final  value  {x)  of  the  time,  in  the  form  [x  —  Xq),  its  sign  is  directly 
reversed  by  the  mutual  interchange  of  the  initial  and  final  values, 
which  transforms  its  expression  to  (.Tq  —  .r). 

393.  Let  F^  denote  the  form,  which  any  function  F  of  the 
final  and  initial  values  of  the  variables  assumes  after  the  mutual 
interchange  of  these  values  ;  and  let 

represent  the  system  of  integral  equations  reduced  so  that  the 
functions  {(fi)  do  not  involve  the  initial  values  (>t'f).  The  inter- 
change of  the  initial  and  final  values  in  this  system,  produces  a 
system  of  integral  equations  in  which  each  variable  is  expressed  in 
terms  of  that  one  variable,  which  represents  the  time,  and  of  the 
arbitrary  constants  which  are  the  initial  values  of  the  variables. 
This  new  system  is  represented  by 

394.  The  discussion  has,  hitherto,  been  limited  to  differential 
equations  of  the  first  order,  but  it  can,  readily,  be  extended  so  as  to 

27 


—  210  — 

embrace  those  of  higher  orders.     If,  for  instance,  the  equations  are 
friven  in  the  form 


o 


in  which  the  functions  (Xj)  may  involve  all  the  derivatives  of  the 
variables  (.r,),  which  are  of  an  order  inferior  to  {pi),  each  of  these 
inferior  derivatives  may  be  regarded  as  an  independent  variable, 
exj)ressed  by  the  form 

With  this  new  system  of  variables  the  given  equations  are 
replaced  by  the  differential  equations  of  the  first  order,  represented 

The  number  of  these  differential  equations  of  the  first  order  is 
easily  seen  to  be  {^iPi  -|-  1)  • 

395.  When  the  differential  equations  are  not  given  in  the 
normal  form  (2IO3),  they  can  always  be  reduced  to  this  form.  For 
this  purpose,  each  of  the  equations,  which  contains  none  of  the 
highest  derivatives  of  the  variables,  must  be  differentiated  as  many 
times,  denoted  by  «•,  as  are  necessary  to  raise  it  to  an  order,  wdiich 
contains  such  derivatives.    If  the  given  equations  are  represented  by 

^  =  0; 

the  equations,  which  are  thus  derived  from  them,  may  be  expressed 


—  211  — 

in  which  «,  is  zero,  when  it  is  applied  to  an  equation  which  is  not 
difFerentiatecl,  Each  of  the  derived  equations  contains  at  least  one 
of  the  highest  derivatives  of  the  variables,  which  may  be  expressed 
bj  Ift'^^-iXi.  The  functions  (9:,)  should  be  independent  functions  of 
these  derivatives ;  whenever  this  is  not  the  case,  such  derivatives 
can  be  eliminated  from  the  derived  equations,  and  one  or  more 
resulting  equations  will  be  obtained  in  which  they  are  not  involved. 
The  independence  of  the  functions  (9^,)  can,  however,  be  directly 
tested  by  means  of  their  determinant  (18029),  which  vanishes  when 
it  is  taken  with  respect  to  quantities,  for  which  these  functions  are 
not  independent. 

When  the  fwiciions  (9,)  are  independent  wiih  resjject  to  the  highest 
derivatives  contained  in  them,  the  reqidred  normal  equations  (2IO3)  ^-^^'^ 
obtained  from  the  given  equations  and  their  successive  derivatives  of  an  order 
not  higher  than  those  of  the  derived  equations  (2IO31)  Ig  the  usual  process 
of  elimination.     For, 

First,  there  is  a  sufficient  number  of  equations,  because  the 
number  of  equations,  added  to  the  given  equations  by  differentiation, 
is  2^i0.i  Avhich  is  the  same  with  the  number  of  derivatives,  superior 
to  the  order  (/>,),  the  highest  of  which  are  to  be  retained  in  the 
normal  equations. 

Secondlg,  these  equations  are  independent  of  each  other  in 
respect  to  the  derivatives  of  the  order  {pi),  and  of  the  superior  orders, 
and,  therefore,  sufficient  for  the  required  elimination;  because  if  any 
of  the  equations  of  the  inferior  orders  were  not  independent,  their 
derivatives,  which  are  included  in  the  group,  (2IO31)  would  not  be 
independent  of  each  other. 

396.  When  the  functions  (9)^)  are  not  independent  with 
respect  to  the  highest  derivatives  contained  in  them,  each  of  the 
equations  of  an  inferior  order,  obtained  from  the  derived  equations 
by  elimination,  can  be  substituted  for  one  of  the  derived  equations, 


—  212  — 

which  is  necessarily  involved  in  the  elimination  by  which  the 
reduced  equation  is  obtained.  If,  therefore,  one  of  the  given 
equations  is  involved  in  the  elimination,  the  order  of  the  given 
equations  is  reduced  by  the  substitution  of  the  given  equation.  But 
if  all  the  equations,  necessarily  involved  in  the  elimination,  were 
derived  by  differentiation  from  the  given  equations ;  and  if  a 
denotes  the  smallest  number  of  successive  differentiations,  by  which 
either  of  these  derived  equations  was  obtained  ;  the  reduced  equa- 
tion is  obviously  a  derivative  of  the  order  {a)  of  an  equation, 
which  can  be  obtained  by  direct  elimination  from  those  of  the 
given  equations,  which  are  of  an  order  inferior  by  (a)  to  the 
derived  equations,  combined  with  the  derivatives  of  the  other 
given  equations  of  an  inferior  order.  This  reduced  equation  of 
an  inferior  order  may,  then,  be  substituted  for  either  of  the  given 
equations  of  a  higher  order,  ujoon  which  its  elimination  neces- 
sarily depends.  In  all  cases,  therefore,  in  ivJiich  the  functions  ((fi)  are 
not  independent  with  respect  to  the  highest  derivatives  contained  in  them, 
the  order  of  the  given  equations  can  he  reduced  hg  the  substitution  of 
an  equation  of  an  inferior  order  obtained  bg  elimination  between  some  of 
the  given  equatiom  and  the  derivatives  of  others,  2vhich  are  of  an  inferior 
order. 

397.  That  the  normal  forms,  obtained  by  the  process  of  §  395, 
are,  as  it  was  remarked  by  Jacobi,  those  which  are  obtained  with  the 
least  complexity  of  operation,  is  easily  perceived  without  any 
attempt  at  demonstration.  It  is,  also,  obvious,  by  what  modes  of 
substitution  other  normal  forms  can  be  derived  from  these,  which 
are  equivalent  to  them  in  the  aggregate  order  of  differentiation,  but 
differ  in  the  distribution  of  the  derivatives.  Thus  if  either  of  the 
functions  (X,)  is  of  an  order  inferior  by  {q^)  to  that  of  the  given 
equations,  it  is  by  (5',)  successive  differentiations  elevated  to  an  order 
which  contains  one   or  more  of   the  highest    derivatives   involved 


—  213  — 

in  the  normal  forms.  The  (5',:)th  derivative  of  the  equation  (2IO3), 
after  the  values  of  the  highest  derivatives,  given  by  the  normal 
equations,  are  substituted  in  its  second  member,  so  that  it  is 
expressed  in  the  form 

may  take  the  place  of  this  equation  in  the  system  of  normal  equa- 
tions. If  then  DY~'^i^i'  is  one  of  the  derivatives  contained  in  {Xi), 
and  if  the  normal  equation  (2IO3)  is  reduced  to  the  form 

])lv-'^iX,'  =  X[', 

it  may  take  the  place  of  the  equation 

D^^'x,r=-X,. 

in  the  group  of  normal  equations.  By  means  of  ( 21814)  and  its 
derivatives  of  an  order  inferior  to  the  (*/,)  th,  all  the  other  equations 
may  be  reduced  so  as  only  to  contain  derivatives  of  {xi/)  of  an  order 
inferior  to  the  [pi,  —  !7,)th.  The  nominal  system  is  hj  iliis  means  trans- 
formed to  another  normal  system,  in  ivhich  the  highest  derivative  of  one  of  the 
variables  is  increased,  just  as  much  as  that  of  another  of  the  variables  is 
decreased. 

398.  The  repetition  of  the  process  of  the  preceding  section 
may  be  so  conducted  that  one  or  more  of  the  variables  shall  finally 
disappear  from  the  system  of  normal  equations,  and  the  number  of 
equations  will  be  simultaneously  diminished  to  the  same  amount  as 
that  of  the  variables.  The  process  may  be  continued,  indeed,  until 
only  two  variables  remain,  one  of  which  is  the  variable  (/),  with 
respect  to  which  the  derivatives  are  taken ;  but  the  reduction  to 
this  form  involves  the  greatest  prolixity  and  complexity  of  computa- 
tion.    There  are  special  cases,  however,  and  particularly  that  of 


—  214  — 

linear  differential    equations,  in  wliicli  this  mode   of  reduction   is 
peculiarly  advantageous. 

The  principal  portion  of  this  discussion  of  differential  equations 
is  the  combined  result  of  the  investigations  of  Euler,  Lagrange, 
Cauchy,  and  Jacobi  ;  but  an  important  addition  to  these  researches 
is  now  to  be  developed,  for  which  geometry  is  eminently  indebted 
to  Jacobi. 


THE   JACOBIAN    MULTIPLIER    OF   DIFFERENTIAL    EQUATIONS. 

399.  The  function,  which  was  called  by  Jacobi  the  neiv  multiplier, 
in  order  to  distinguish  it  from  the  Eulerian  multiplier,  but  which,  on 
account  of  its  superior  importance,  is  here  distinguished  simply  as 
the  multiplier  of  a  linear  partial  differential  equation  of  the  first  order 
represented  by  (2OO2),  is  that  function  ivhich,  miiltipjlied  hj  this  equation, 
renders  its  first  member  an  exact  functional  determinant  ( "Sl^)  of  the  indefi- 
nite function  (/)  and  of  n  undefined  functions  (f)  ivith  respect  to  the  (??  -j-  1) 
variables  [xi),  ivhich  are  the  independent  variables  of  the  given  equation.  On 
account  of  the  mutual  relations  of  the  partial  differential  equation 
(2OO2)  and  the  simultaneous  differential  equations  (199i2),  this  same 
function  may  also  be  regarded  as  a  multiplier  of  the  differential  equations 
(199i2)  ;  and,  for  the  same  reason,  it  may  be  considered  as  a  multiplier 
of  the  linear  partial  differ eiitial  equation  of  the  first  order  (2OO20)  of  n 
independent  variables. 

400.  If  either  of  the  functions  (/,),  or  any  function  of  these 
functions,  is  substituted  for  /,  the  determinant  vanishes,  by  §  352, 
and  the  equation  (2OO2)  is  satisfied.  The  functions  (f)  are,  therefore, 
n  independent  solutions  of  the  equation  (2OO2). 

401.  If  the  multiplier  of  the  equation  (2OO2)  is  denoted  by 
^'(0,  the  condition,  by  which  the  multiplier  is  defined,  is  expressed  by 


—  215  — 
the  identical  equation 

The  equality  of  the  coefficients  of  Dx.f  in  the  two  members  of 
this  identity  is,  by  the  notation  adopted  in  the  theory  of  determi- 
nants, expressed  by  the  formula 

The  substitution  of  this  value  of  ok^  in  the  equation  (ISOa)  gives 
the  equation 

which  is  a  linear  2^(^}'ticil  dljf'ercnikd  equation  of  Ihe  jird  order,  l>j  ivhivh 
the  multiplier  is  analytically  defined. 

402.  The  defining  equation  of  the  multijDlier  may  by  (lOOia) 
be  developed  into  the  form 

:e,{x,d.^^{^  +  ^K^D.x;)  =  :e,{d,m^d,^  +  ^^d^^x^  =  o, 

or 

This  equation  divided  by  ^jait  becomes 

r.log  ^((d  +  ^,D,X,  =  D  log  .d(o  +  2:,D.X,  ■=  0. 

If  all  the  A'^ariables  are  regarded  as  functions  of  .r,  and  if  x  is 
introduced  in  place  of  the  element  of  variation,  by  means  of  the 
formula 

Dx  =  X, 


—  216  — 
the  preceding  equation  finally  assumes  the  form 
XD,  log  ^tD  +  :E, Z>.^.X,  =  0 ; 

which  is  an  equation  involving  common  differentials,  hj  ivJiich  the  miiUiplier 
is  analf/ticallf/  defined. 

403.  The  equation  (2158)  gives,  by  (lOlg),  when 

z  =  0, 
the  value  of  the  multiplier  in  the  form 

404.  If  the  values  of  (/)  are  expressed  in  terms  of  (x,),  by 
means  of  the  equations  (ISOja),  and  if,  by  reason  of  the  integrals 
(199io),  the  constants  {a^  are  substituted  for  (/),  the  value  of  the 
multiplier  becomes 

^'""  —  y      )   X' 2:±I)a,F,Da,F, Ba,F^' 

in  which  the  sign  may  be  rejected  at  pleasure. 

405.  In  the  particular  case,  in  which  the  equations  (189i2) 
assume  the  form 

in  which  the  functions  (cj),)  involve  the  arbitrary  constants  («,), 
together  with  no  other  variable  than  x,  the  value  of  the  multiplier 
is  by  (ISOia)  reduced  to 

^t  =  — _— 1 1 

XZ±Da^%Da,% DaJP,,  ~  X2^+  Da,X,Da,X, Da,X,, 

__  1  _  1 

X2:±Df,x^I)j,x, J)f,^x,,  —  X^}j\,J 


—  217  — 

which  equdtlon  might  have  been  directly  deduced  from  (216ii)  and 
(187io). 

406.  If  the  functions  (i^)  are  given  independent  functions  of 
(/j),  they  are  independent  solutions  of  the  equation  (2OO2)  ^^^  give 
a  multiplier  (^ii^)  different  from  '^^  and  which  is  determined  by 
the  equation  derived  from  (216n), 

X^i^i  =  2:±  D.^F^D^^F^ D.,F,,. 

This  equation,  by  means  of  (I8614)  and  (216n),  assumes  the  form 

X^{K=^,^^2±Df^F,Df^F, Df^F^ 

=  X^i^  ^±Df^F,Df.^F^ Df,F^, 

which  gives 

!^=  V  +  Dy,F,Df,F, Dr^F,. 

The  second  member  of  this  equation  is  a  function  of  the  func- 
tions (/,),  and  may  be  an  arbitrary  function  of  these  functions,  so 
that  it  can  have  n  independent  values.  The  equation,  therefore, 
serves  to  determine  n  -\-\  independent  values  of  the  multiplier 
(vj^Xfo^),  which  is,  by  (215i2),  the  whole  number  of  independent  values 
of  Avhich  it  is  susceptible.  Hence,  the  ratio  of  any  two  multipliers  is  a 
solution  of  the  equation  (2OO2).  It  also  follows  from  this  argument  that 
every  solution  of  the  equation  (215i2)  is  a  value  of  the  multiplier. 

407.  In  the  particular  case,  in  which 

Z,D.,Xi^^, 

one  of  the  n  -\-  1  solutions  of  (215i2)  is  reduced  to  a  constant,  so  that 
in  this  case,  the  constant  must,  contrary  to  the  ordinary  usage,  be 
included  among  the  solutions  of  the  equation.     The  constant  may 

28 


—  218  — 

be  supposed  to  be  unity,  and,  therefore,  one  of  the  muMpUers  of  the  equa- 
tion (2OO2)  is  unity,  tvhcn  the  condition  (2I727)  2^  fulfilled,  and  all  the  other 
multipliers  are  solutions  of  the  equation  (2OO2). 

408.  When  the  solutions  (/)  of  the  equation  (2OO2)  are  known, 
the  corresponding  value  of  the  multiplier  may  be  determined  from 
(2I611).  But  it  can  be  derived  by  a  shorter  process,  when  either 
of  the  solutions  (aa(ti)  of  (215i2)  is  known,  and  also  the  initial  value 
of  q4\  Thus  if  IT  denotes  the  ratio  of  'Jatt.  to  <£>Ms>,  the  equation 
(216n)  gives  by  (194,), 

When  the  initial  values  are  substituted  in  this  equation  with 
the  notation  of  §  393,  it  becomes 


ok' 

The  value  of  77°  may,  by  the  elimination  of  the  variables  {xf) 
be  reduced  to  a  function  of  the  functions  [fi) ;  and,  if  in  this 
expression  the  functions  {f)  are  substituted  for  their  initial  values 
(/J),  the  value  of  77  is  reproduced.  For  the  function,  which  is 
obtained  by  this  substitution,  is  a  function  of  {f)  and  therefore  a 
solution  of  the  equation  (2OO2);  and  it  is,  moreover,  that  particular 
solution,  of  which  the  initial  value  is  the  given  function  77°. 

409.  In  the  especial  case,  in  which  the  initial  values  of  [f]  are 
the  variables  {x^),  the  value  of  Qk°  is  obviously  reduced  to  unity  and 
the  equation  (218ii)  becomes 

77°=vfc(i:X°. 

410.  When,  in  the  differential  equations  (199i2),  the  arbitrary 


—  219  — 

element  of  variation  is  assumed  to  be  the  variable  x,  the  value  of  X 
is  unity ;  and,  in  this  case,  the  equation  (218ii)  becomes 

ok; ' 

which  in  the  case  of  the  preceding  section  is  reduced  to 

and  when,  moreover,  the  equation  (217^7)  '^^  satisfied,  so  that  one  of 
the  multipliers  is  unity,  this  value  is  still  further  reduced  to 

17°  =1. 

411.  The  arbitrary  constants  («,)  may  be  substituted  for  the 
functions  (/,)  in  the  equation  (218n),  when  it  is  regarded  as  result- 
ing from  the  integrals  of  (lOOia).  By  this  substitution  U  becomes  a 
function  of  the  arbitrary  constants,  which  may  be  represented  by  C, 
and  the  equation  gives,  by  means  of  (187io), 

^l,n  =  -^  i  Da^XiDa^X^ Da^X^  =  ^k  ■  X' 

The  logarithm  of  this  equation  becomes  by  the  substitution  of 
(2I62),  and  including  C  in  the  constants  of  integration, 

log  -S-  +  I)a,X^  I)a,X, Da„X,  =  log^  +  log   C—  log  <^i^, 

in  which  all  the  functions  (X,)  can  evidently  be  multiplied  by  any 
common  factor,  without  disturbing  the  equality. 

412.  In  the  especial  case  of 

X=l 


—  220  — 
the  preceding  formula  becomes 

413.  When  simultaneous  differential  equations  are  transformed 
from  one  system  of  variables  to  another,  the  multiplier  usually  under- 
goes a  change  at  the  same  time,  but  there  are  conditions,  to  which 
the  arbitrary  element  of  differentiation  may  be  subjected,  and  under 
which  the  multiplier  remains  unchanged.  Thus  if  the  new  system 
of  variables  is  represented  by  (tVi),  if  the  equations  (199i2),  in  their 
new  form,  are  represented  by 

in  which  the  accented   sign   of  differentiation   refers   to    the    new 
arbitrary  element  of  differentiation,  and  if 

-jj  —  ^y 

the  values  of  ( Tl^)  become,  by  (lOOa^)  and  the  preceding  formulae  of 
this  section. 

This  value  of  ( TF^),  in  combination  with  the  formulae  (lOOgs)  and 
(2162),  gives 

^G^, {X.D.J) 


—  221  — 

If  oN*  is  a  multiplier  of  (22O12),  the  defining  equation  of  (2102) 
is,  in  respect  to  this  multiplier, 

cN'^,(  W.D.J)  =  2  +  DJD..J, i?.„/„. 

The  ratio  of  the  equations  (22O27)  ^"^^  (22I3),  reduced  by  means 
of  (I8613)  and  (187io),  gives 

^  oir  _2±DJD.J, ZJ.„X 

=  -2"  +  D^^xDxr^Xi Du-^x^ 

=  {^±  DjvD.^iv, D^w,,)-\ 

If,  therefore,  the  multipliers  oV  and  <£A^  are  equal,  the  value  of 
G  becomes  G',  if 

G'  =^  ^  +  D^,xDw^Xi Dii-n^n 

=  {:E±  D,ioD.^iL\ D.w:)-^. 

414.  The  equation  (21525),  applied  to  the  new  system  of  varia- 
bles (?6',),  gives,  by  means  of  this  equation  and  (22O17),  if  the  multi- 
pliers are,  for  the  instant,  assumed  to  be  equal, 

^,Z>„.  TT^  =  —  i>'  log  ^(t  =  —  6^'i>  log  ^tt 

=  g':e,d.^x,. 

415.  If  the  arbitrary  element  of  differentiation  is  supposed  to 
be  the  same  in  both  systems  of  variables,  the  values  of  G,  Wi,  and 
oN*  become 

G=l, 

c^=G'^Ais. 


—  222  — 

416.  If  the  first  m  -f- 1,  only,  of  the  variables  (:r,)  are 
exchanged  for  the  new  variables  {u'i),  which  limitation  is  expressed 
by  the  formula 

the  value  of  G^  is  abbreviated  to 

G'  :=:S± D^xDu^^Xi -Z>«.,„rr^ 

=  {2±D^wD.^iv, i>.„.«^.)-^ 

417.  Hence  if  the  arbitrary  element  of  differentiation,  com- 
mon to  the  two  systems,  is  one  of  the  variables  and  is  expressed  by 
t,  so  that  the  remaining  variables  are  still  denoted  by  (x^)  and  (tt\)y 
the  formula  (22I15)  continues  to  express  the  value  of  G\ 

418.  If  the  last  {n  —  m)  of  the  variables  {n\)  are  solutions  of 
the  equation  (2OO2),  the  corresponding  values  of  the  functions  ( TF|) 
vanish  by  (22O22).  If  the  multiplier  is  also  supposed  to  remain 
unchanged,  the  partial  differential  equation  (2OO2),  by  which  it  is 
determined,  is  reduced  to 


'J 


0 

The  arbitrary  constants  (/:?,)  may,  therefore,  be  substituted  for 
the  solutions  (?t',),  and  the  value  of  G'  becomes 

419.  But  if,  instead  of  the  equality  of  multipliers,  the  ele- 
ments of  differentiation  are  identical  in  the  systems,  the  defining 
equation  is  expressed  in  the  slightly  different  form  of 


—  223  — 

in  which  the  functions  ( TT',)  and  the  muUipher  (oS')  are  given  by 

(221»). 

420.  If  the  variables  (tVi)  which  are  retained,  coincide  with 
the  original  variables  (x^),  the  equation  for  the  multiplier  becomes 

m 
0 

in  Avhich 

By  the  formulae  of  this  and  the  two  preceding  sections  the 
multiplier  of  the  system  of  differential  equations,  to  which  a  given 
system  is  reduced  by  means  of  any  of  its  integrals,  can  be  obtained 
from  the  multiplier  of  the  given  system.  This  will,  soon,  appear  to 
be  one  of  the  most  important  properties  of  multipliers. 

421.  If  the  given  differential  equations  are  of  an  order,  which 
is  higher  than  the  first  order,  and  have  the  normal  form  (210g),  the 
equation  (21525),  by  which  the  multiplier  is  defined,  is  simplified  by 
the  consideration  that 

> 

The  multiplier  of  the  given  equations,  or  of  the  equations  (2IO15),  ^^ 
which  tliey  should  he  replaced,  is,  therefore,  determined  hy  the  equation 

B  log  ^((d  -f  ^  .Z>,tP -1)  Xi  =  0 . 

422.  If  the  functions  JQ  do  not  involve  a:/^"^'  or  if,  in  general, 

unity  is  one  of  the  values  of  the  multiplier  of  the  given  equations. 


224  

423.  If  the  given  equations  have  not  the  formal  form,  but 
have  the  form 

such  that  they  involve  no  derivatives  of  a  higher  order  than  the  nor- 
mal forms,  to  which  they  are  reducible  by  immediate  elimination 
without  differentiation,  the  equation  for  determining  the  multiplier 
assumes  a  simple  symbolic  form,  by  means  of  the  notation 

For  it  is  to  be  observed  that  each  of  the  subsidiary  terms,  of 
which  the  second  term  of  the  equation  (22825)  is  the  aggregate,  is  to 
be  obtained  from  the  equations  (2243),  by  taking  their  derivatives 
relatively  to  a^^r'^^  on  the  hypothesis  that  a^^k*  are  functions  of  this 
variable,  and  thence  determining,  by  elimination,  the  values  of  these 
subsidiary  terms.     Hence  if 

the  derivatives  of  (2243),  relatively  to  x^l'r'^^  are  represented  by 
( 17724),  provided  the  letters  t  of  that  equation  are  accented  i  times, 
and  the  number  k  is  written  below  the  u.  From  the  comparison  of 
(ISOig)  with  (224ii),  it  appears  that  {i,}c)  vanishes  in  the  present  case, 
and  that  the  sign  of  d  is  to  be  reversed,  whence  the  equation  (I8O26) 
becomes 

The  equation  (22825)  ^^  which  the  multiplier  is  determined,  assumes  the 
symholical  form 

D  log  ^ti  =  _  :^,m  —  (T  log  'Sto^. 


—  225  — 

424.  It  may,  sometimes,  happen  that  the  vakies  of  a^[^  and 
^.«i,'*  are  such  that  the  sum  of  da^l\  and  of  lDa^/^\  in  which  I  is 
constant,  is  simpler  than  da^^K     In  this  case,  if 

d'^d  +  in, 

the  addition  of 

to  the  equation  (2243i)  gives  the  sf/mhoUcal  form 
i>log(.ci(tQj5^j^)  =  (5qog^.,. 

425.  If  the  given  differential  equations  have  the  form  (2IO27), 
so  that  they  cannot  be  reduced  to  the  normal  form  without  differ- 
entiation, the  equations  (2IO31),  wdiich  are  derived  from  them  by 
differentiation,  give,  by  direct  elimination,  a  system  of  normal  forms, 
which  include,  as  a  reduced  system,  the  normal  forms  finally  obtained 
by  the  process  of  §  395.  The  multiplier  of  the  equations  (2IO27)  is 
determined  by  the  symbolic  equation  (22*43i),  or  (225io),  provided 
that  in  the  values  (224io)  of  a^^  and  da^l'^  from  which  ()^'3l„  is  consti- 
tuted, the  value  of /?^.  is  increased  by  a^. 

426.  The  values  of  «^'  and  (5'«SJ*  may  be  determined  directly 
from  the  equations  (2IO27).  For  this  purpose,  if  X  is  written  instead 
of  a  in  order  to  avoid  the  confusion  which  might  arise  from  the  use 
of  a  as  an  arbitrary  constant,  and  if  the  ingenious  notation,  which 
is  familiar  to  the  German  mathematicians,  for  the  continued  product 
of  all  the  integers  from  1  to  X  inclusive. 


"O" 


l\  =  l{l  —  l){l^2) 3.2.1^ 

is  adopted,  the  equations  (2IO31)  are  represented  by 

29 


—  226  — 
and  we  find,  by  well-known  formula?, 

D^^K)  9)  =  1"^  D}  {p^M  FD,(K)  :?;''■'] 

V  ' 

,/'L(x-r)!(;.-.  +  ,.)!       '  ^  J 

The  inferior  limit  v  is  determined  by  the  condition  that  neither 
V  nor  X  —  yi  A^  v'  can  be  negative.     Hence 

if  X  +  1>K,  /==0, 

if  ^_l<x,  /=:x  — ;.. 

In  the  former  of  these  two  cases  the  last  term  is 

but  in  the  latter  case  it  is  simply 

D^^k-dF. 

It  follows,  then,  from  (224]o)  that,  since  i^,  does  not  contain  any 
higher  derivative  of  x^.  than  jh^ 

daf^D^p^^)F,  +  hD,af. 

427.  The  system  of  normal  equations,  derived  by  the  process 
of  §  395,  is  related  to  the  system  of  normal  forms,  which  has  been 
discussed  in  the  preceding  sections,  precisely  as  any  reduced  system 
of  differential  equations  is  related  to  that  from  which  it  is  reduced 
by  means  of  a  portion  of  its  integral  equations.  The  integral  equa- 
tions are,  in  this  case,  the  equations  (2IO27)  and  all  their  derivatives, 


which  are  inferior  to  the  final  derivatives  expressed  by  equa- 
tions (210iji),  the  multiplier  of  the  reduced  equations  is,  conse- 
quently, obtained  by  dividing  the  multiplier  aLlb  of  the  equations 
(2IO31)  by  the  function  G  given  by  the  expression  (2289).  The 
functions  (iv),  involved  in  the  value  of  G,  represent  the  first  mem- 
bers of  the  integral  equations  (2IO27)  '^^^^  their  derivatives.  But  it 
follows  from  (226,)  and  (22623)  that 

The  equations  (2IO27)  '^'^^y^  now,  be  supposed  to  be  arranged  in 
an  order  conformable  to  the  orders  of  the  derivatives,  by  which  they 
are  brought  to  the  form  (2IO31),  so  that  those,  of  which  the  higher 
orders  of  derivative  are  taken,  may  precede  the  equations  of  which 
lower  orders  are  taken.  Instead  of  reducing  the  equations,  by  a 
single  step,  to  the  final  system,  the  reduction  may  be  accomplished 
by  successive  steps ;  and,  at  each  step,  the  derivatives  of  the  equa- 
tions (2IO2-),  which  are  admitted  into  the  group  of  integrals,  may  be 
diminished  by  unity,  while  the  number  of  accents  of  the  eliminated 
variables  is  also  diminished  by  unity.  At  the  step  denoted  by  h, 
therefore,  the  derivatives  of  those  equations  (2IO27)  ^^^  added  to  the 
group  of  integrals  for  which  the  orders  of  derivative  (?.,)  are  greater 
than  h.  At  this  step  a  factor  [Gi^)  of  G  is  also  obtained,  and  all  the 
derivatives  of  which  it  is  composed  are  represented  by  the  functions 
(«i.''),  in  which  the  superior  limit  of  k  is  the  same  with  that  of  /. 
Hence  if 

h  —  h+i>0, 
the  value  of  the  factor  of  G  is 


—  228  — 
but  if 

this  factor  is 

The  logarithm  of  the  complete  value  of  G  is,  therefore, 

log  6^=:^.  [(?,  +  !-?,)  log  ^S]. 
PRINCirLE    OF    THE    LAST    ML'LTIPLIER. 

428.  The  consideration  of  the  case  in  which  there  are  two 
variables,  leads  to  a  valuable  principle  of  integration,  discovered  by 
Jacobi,  and  which  he  called  the  principle  of  the  last  multiplier.  In  the 
case  of  two  variables,  the  equation  (2162)  becomes 

which  gives 

Hence  it  is  obvious  that 

Df,  =  ^(t  {XDx  —  X,Dx\) 
or,  by  integration. 


/i=   C^{\^{XDx  —  X^Dx^), 


so  that  ivhen  the  multiplier  is  hioivn,  this  equation  determines  the  integral  of 
the  ttvo  differential  equations  (199i2)  of  tivo  variables,  or  that  of  the  sin- 


—  229  — 
gle  equation  to  which  they  are  eqidvaleiity 

and  the  miiUijjUcr  is,  in  this  case,  identical  iviih  the  tvell-hiown  Eulenan 
multiplier. 

429.  ^Yhcn  all  the  integrals  hut  one  of  a  given  sgstem  of  dij^erential 
equations  (199i2)  are  hiown,  of  ivhich  the  multiplier  is  also  given,  the  last 
integral  is  determined  hy  quadratures  ly  the  ^^rocess  of  the  preceding  section  ; 
because  the  multiplier  of  the  two  differential  equations  with  two 
variables,  to  which  the  given  system  may,  in  this  case,  be  reduced,  is 
determined  from  the  given  multiplier  by  §  418.  This  is  Jacobi's 
principle  of  the  last  multiplier. 

430.  In  the  case  of  §  380,  in  ivhich  the  element  of  variation  [t)  is  one 
of  the  variables,  if  the  functions  {Xi)  do  not  involve  it),  the  equation  (2018) 
gives 

i 

from  which  t  can  be  determined  by  quadratures,  when  all  the  other 
integrals  of  the  given  equations  are  known,  even  if  the  multiplier  is 
not  known,  provided  that  Xj  is  reduced  to  a  function  oi x^,  by  means 
of  the  known  integrals. 

If  the  multiplier  is  also  Jcnown,  and  if  it  does  not  involve  t,  the  last  of 
the  integrals  ivhich  do  not  involve  t  can  he  determined  hy  the  process  of  the 
preceding  section,  and,  therefore,  the  two  last  integrals  of  the  given  equations 
can,  in  this  case,  he  determined  hy  quadratures. 

Bid  if  the  given  multiplier  (^(t)  involves  t,  a  mxdiiplier  {ysMs^^,  ivhich 
does  not  involve  t,  can  he  derived  from  all  the  integrals  ivhich  do  not  involve 
t,  and  the  quotient  of  these  two  multipliers  gives  hy  §  406,  an  integral  involv- 
ing t,  and  ivhich  taJces  the  place  of  (229i7) ;  ^^  ^^^^<^?  ^'^^  ^-^"'^  ^^5^,  the  last 
integral  is  determined  in  a  finite  form  without  integration. 


—  230  — 

431.  This  proposition  was  shown  by  Jacobi  to  admit  of  the 
following  generalization.  If  all  the  functions  {X^),  in  tvhich  i  is  greater 
than  m,  are  free  from  those  of  the  variables  [xi)  in  ivhich  i  is  not  greater 
than  m,  and  if  the  remaining  functions  satisfy  the  equation 

m 

0  * 

two  integrations  can  always  he  performed  hy  quadratures,  whenever  a  multi- 
plier is  hioivn  tvhich  does  not  involve  the  variables  (:?^,<^  +  i),  hut  when  the 
given  multiplier  does  involve  either  of  these  variables  one  integration  can  he 
performed  hy  quadratures,  and  another  integral  is  given,  immediately,  with- 
out any  process  of  integration.  For  if  the  given  multiplier  <£J^  involves 
only  the  variables  (.^^>^),  it  not  only  satisfies  the  condition  (215g), 
but  also  on  account  of  the  equation  (2306) 

and  is,  therefore,  a  multiplier  of  the  portion  of  the  equations  (199i2) 
in  which  i  is  greater  than  m.  This  portion  of  the  given  equations 
can,  therefore,  be  first  integrated,  independently  of  the  remainder 
of  the  system,  and  the  last  integral  of  this  portion  will  be  obtained 
by  quadratures,  because  its  multiplier  is  given.  But  the  last  inte- 
gral of  the  whole  system  may,  also,  be  obtained  by  quadratures, 
because  its  multiplier  is  known;  so  that  two  of  the  integrals  can  be 
obtained  by  quadratures. 

But  if  the  given  multiplier  involves  any  of  the  variables 
(^i<»n  +  i)>  the  separate  integration  of  that  portion  of  the  equations 
(199i2)  in  which  i  is  greater  than  m,  gives  a  multiplier  of  this  portion 
involving  only  the  variables  (.'r,^,^),  which  satisfies  the  equation  (23O15)  J 
and  by  (2306)  it  also  satisfies  the  equation  (2158),  so  that  it  is  a  new 
multiplier  of  the  given  equation.     The  quotients  of  these  two  mul- 


—  231  — 

tipliers  gives,  by  §406,  an  integral  involving  (.r,<-,„^i),and  which  takes 
the  place  of  the  first  of  the  two  integrals,  which  are  obtained  by  quad- 
ratures when  the  given  multiplier  involves  only  the  variables  (:i\>,„). 


PARTIAL     MULTIPLIERS. 

432.  Additional  to  the  systems  of  Eulerian  and  Jacobian  mul- 
tipliers, and  inclusive  of  them,  are  those,  of  which  I  have  given  the 
investigation  in  Gould's  Astronomical  Journal,  and  which  I  have  called 
partial  inuliipliers.     The  partial  multipliers  of  the  differential  equations 

(199i2)  are  represented  by  (^si^il^i^,,^,,^^ ^.J,  in  which  i,ki,k2,. . .  etc. 

are  any  different  numbers,  or  by  (^'^'f^/,A-)j  ii^  which  I  and  K  denote 
groups  of  numbers ;    and  they  are  defined  by  the  equation 

f<£i^^j=:e±d  ad  f, DA, 

in  which  P  is  any  arbitrary  function,  /"i ,  /"a  •  •  •  hi  are  numbers  not 
included  in  the  groups  I,  and  /u/a,  etc.  are  solutions  of  the  equa- 
tion (2OO2).  The  notation  (o&lU^')  may  also  be  used  to  denote  the 
multiplier,  with  the  definition  that  if 

^denotes  the  group  of  numbers  represented  by  (/",„). 

433.  The  system  of  multipliers  of  (lOOja),  evidently,  satisfies 
the  system  of  differential  equations,  which  are  derived  from  (187io), 
and  represented  by 

in  which  i  includes  all  the  numbers  not  belonging  to  the  group  I. 

434.  The  group  of  all  the  numbers  not  included  in  the  group 


—  232  — 

(/)  with  the  exception  of  any  two,  which  may  be  selected  at  pleas- 
ure, may  be  denoted  by  H.  The  elimination  of  the  corresponding 
values  of  X/.  from  the  equations,  obtained  from  (2OO2)  by  the  substi- 
tution of  the  various  values  of  (/)  gives  the  equations,  which  are 
represented  by 

This  system  of  equations  combined  with  that  of  (23l2s)  defines, 
analytically,  the  sj^stem  of  partial  multipliers. 

435.  In  the  formation  of  the  multipliers,  a  careful  regard  must 
be  had  to  their  signs,  conformably  to  the  rule  of  formation  of  deter- 
minants, so  that  in  general 

436.  In  the  special  case,  in  which  the  group  (/, /)  of  §433 
is  reduced  to  a  single  number,  and  in  which  P  is  X,  the  preceding 
equations  become 

0  =  ^,D^^  (X^((o,)  =  :^,D^^  (a:,(t  X,) ; 

so  that,  the  miiUij)l'ier  is,  in  this  case,  the  Jacohian  multiplier. 

437.  In  the  case,  in  which  the  groups  {i,I)  of  §  433  include 
the  numbers  of  all  the  variables  but  one,  and  in  which  P  is  unity, 
the  equations  become 

I  k 

so  that,  the  system  of  multipliers  is,  in  this  case,  that  of  the  Eulerian  multi- 
pliers amplified  hy  Lagrange. 


—  233  — 

43S.  The  partial  multipliers  may  be  denoted  as  ihc  frsf,  second, 
etc.,  to  the  lad  corresponding  to  the  degree  of  the  determinant  which 
is  the  second  member  of  the  equation  (216n)-  With  this  designa- 
tion, the  last  multiplier  coincides  with  the  Jacobian  multiplier  and 
gives  a  last  integral  of  the  differential  equations,  while  the  first  mul- 
tipliers coincide  with  the  Eulerian,  of  which  each  system  gives  a 
first  integral  of  those  equations.  This  proposition  may  be  general- 
ized, and  it  may  be  shown  that  each  Si/stem  of  muUijjliers  determines 
an  integral  of  the  given  equations  ly  means  of  quadratures,  and  holds  a  place 
in  the  rank  of  multipliers  similar  to  that  held  hy  the  integral,  in  the  rank  of 
{integrals. 

The  investigation  of  the  relations  of  the  multipliers  of  differ- 
ent systems  will  be  found  to  lead  immediately  to  this  proposition, 
after  its  truth  has  been  established  in  the  case  of  the  Eulerian  mul- 
tipliers. 

439.  The  deduction  of  an  integral  of  a  system  of  differential 
equations  (199i2),  by  means  of  quadratures,  from  a  given  system  of 
Eulerian  multipliers,  is  quite  a  simple  process.  For  the  definition  of 
these  multipliers  in  §  437  gives 

Df^^,UK\^^Dx^. 

If  the  quantities  represented  by  ( Q^)  are  defined  by  the  equa- 
tion 

Ox,  '    0 

the  required  integral  is 

f=^,q^  =  a. 

For  the  defining  equation  of  Q^  gives 

'   0 

30 


—  234  — 

Hence  it  is  found  by  differentiation  that  ( Qi)  is  free  from  all 
the  variables  (%<,),  for  if  this  is  supposed  to  be  proved  for  ( Q;,<^i)  it 
it  seen,  by  (23227),  that 

hi  fi  *0  0 

The  differential  of  (23337)  i^?  therefore. 

•  0 

which  corresponds  to  the  required  differential  (233i9). 

440.  When  the  differential  equations  (lOOja)  are  transformed 
to  other  variables  in  the  manner  which  is  indicated  in  §  413,  any 
multiplier  of  the  new  system  is  obtained  by  the  following  formula 
which  corresponds  to  (23I15), 

P'd<^=Z±  D^^f,  D,,^f, D,,J,, . 

If,  then,  the  functions  {G)  are  defined  by  the  equation 

=z{Z  +  D    tvu  D^  IV, D^  tuj,  )-^ 

1  2  'ill    '" 

the  proposition  (I8620)  gives  by  (23I15) 

P^Q>r^=P^^(^^G^(/f)). 

441.  If  any  of  the  solutions  (/)  of  (2OO2)  are  known,  they 
can  be  assumed  as  new  variables  to  take  the  place  of  either  of  the 
given  variables,   and  the  new  multipliers  must  be  determined  by 


—  235  — 

the  preceding  equation.  But  it  is  evident  tliat,  in  this  case,  the 
number  of  elements  which  compose  each  of  the  terms  of  (cV^) 
will  be  diminished  by  a  number  equal  to  that  of  the  solutions,  which 
are  introduced  as  variables.  Hence  since  ?)i  is  the  number  of  ele- 
ments which  compose  each  term  of  (oE^ttj),  if  [m  —  1)  is  that  of  the 
known  solutions,  the  number  of  elements  of  (g-V^)  may  be  reduced 
to  one,  in  which  case  the  multipliers  (g^h)  become  Eulerian  and 
give  the  mih  solution  of  (2OO2)  or  the  mth  integral  of  (IQOja),  by 
means  of  quadratures,  which  corresponds  to  the  proposition  of  §  438. 


III. 

INTEGRALS    OF    THE   DIFFERENTIAL    EQUATIONS    OF   MOTION. 

442.  When  the  differential  equations  of  motion  are  expressed 
in  their  utmost  generality,  there  is  no  known  integral  which  is  suf- 
ficiently comprehensive  to  embrace  them.  But  the  equation  (163i4) 
of  living  forces  is  an  integral,  which  is  applicable  to  all  the  great 
problems  of  jihysics,  and  holds  the  most  important  jDOsition  in  refer- 
ence to  investigations  into  the  phenomena  of  the  material  world. 
There  are  other  integrals  of  great  generality,  which  might  be  inves- 
tigated in  this  place,  if  they  were  not  clothed  with  such  a  character 
of  speciality,  that  they  properly  belong  to  some  of  the  following 
chapters.  The  application  of  Jacobi's  principle  of  the  last  multi- 
plier to  dynamic  equations  gives  results  of  so  general  a  character, 
that  their  investigation  cannot  appropriately  be  reserved  for  any 
chapter  devoted  to  the  consideration  of  special  problems. 


—  236  — 


tiik  application   of  jacobl's   principle   of  the   last  multiplier   to 
Lagrange's  canonical  forms. 

443.  It  follows  from  the  homogeneous  nature  of  T  (165io), 
that  each  of  Lagrange's  equations  (164i2),  involves  one  or  more  of 
the  quantities  represented  by  (>/'),  and  the  system  of  these  equa- 
tions has,  therefore,  the  form  represented  by  (2IO30).  If,  then,  (a^') 
denotes  the  coefficient  of  (i^^)  in  the  value  of  (tu^),  given  by  (IGoj), 
this  value  becomes 

and  that  of  T  is  by  (165n) 

so  that  the  functions  [af)  only  involve  the  quantities  represented  by 
(1])  and  the  time  (/),  and  satisfy  the  equations 

444.  Each  of  Lagrange's  equations  may  be  expressed  in  the 
form 

i  i 

Hence,  when  £1  is  only  a  function  of  (j^,)  and  t,  the  equa- 
tions (224io)  become 

D  rr(fi  —  af, 
k 


_  237  — 
from  which  are  easily  derived  the  equations 


k  i 

The  notation 

k  i 

gives 

[i,k)=  —  {Jc,{), 
dd^  =  D,a^^  +  {i,Jc). 

In  the  substitution  of  these  values  in  (2243i),  it  is  evident  from 
(ISOig),  (I8O31),  and  (ISlg)  that  the  functions  {i,Jc)  disappear,  and 
since  D  takes  the  place  of  i>,,  (2243i)  becomes 

i>logaB.((er=:i>log^^„, 

and,  therefore,  since  the  arbitrary  constant  may  be  neglected, 

which  holds,  even  if  the  equations  of  condition  involve  the  time. 

In  all  dynamical  problems,  therefore,  in  ivhich  the  forces  are  indepen- 
dent of  the  velocities  of  the  moving  hodies,  a  Jacohian  multiplier  is  given 
directly  by  the  equation  (237i9),  so  that  the  last  integral  can  always  be 
obtained  by  quadratures. 

445.  Hence,  by  §  430,  in  any  dynamical  problem,  in  ivhich  the 
forces  and  equations  of  condition  are  independent  of  the  time  as  tvell  as  of 
the  velocities  of  the  bodies,  the  two  last  integrals  can  be  obtained  by  quad- 
ratures. 

446.  The  substitution  of 

hi  =  Xi\J  m,Ui  +  i  —  l/i\l  i^h^'Ui  +  i^'  ^.\l  m„ 


—  238  — 
in  (lG42o)  and  (16228)  gives 

2  2  2 

Hence  if 
the  value  of  w^^'^  is  by  (23G15), 

which,  combined  with  §  §  346  and  348,  gives 

in  which  'S^*/*''  denotes  the  functional  determinant  of  a  group  {M)  of 
{n  -\-  1)  of  the  functions  (^"i)  relatively  to  the  variables  [r]i).  It  may 
be  observed  that  if  n^  is  the  number  of  bodies  of  the  system,  and  n^ 
the  number  of  conditional  equations,  the  value  of  n  is 

«  rrr  3  n^ t^ 1 . 

447.     If  the  conditional  equations  are  represented  by 

and  if 

k 
their  derivatives  with  reference  to  [iji)  are  represented  by 

If  then  (H)  denotes  any  group  of  n  of  the  quantities  (^",),  and 


—  239  — 

{n,h)  denotes  a  group  of  72-(-l  of  the  same  quantities  in  which  the 
group  [H)  is  included,  the  preceding  equations  give,  by  elimination, 
between  all  those  in  which  i  remains  unchanged, 

Since  then  the  group  {H,h)  is  also  denoted  by  (oeXId),  if  the 
group  of  all  the  remaining  quantities  (ij)  is  denoted  by  {N),  '\?  M' 
and  N'  are  other  groups  of  the  same  species,  and  if  ( ^'^^)  denotes 
the  determinant  of  the  corresponding  values  of  (4/')>  ^^^®  preceding 
equations  give,  by  elimination, 

n  n  i 

which,  it  is  easily  seen,  may  be  extended  to  the  case  of  any  groups 
whatever  (iltf  and  Mi),  in  which  each  includes  {n-\-  1)  of  the  quan- 
tities {li).  If,  therefore,  some  one  group  is  arbitrarily  selected  and 
denoted  by  {M^i),  the  equation  (28813)  becomes 

448.     If  the  derivatives  of  (1],)  relatively  to  (^,)  are  denoted  by 

and  if  ^^^^  denotes  the  determinant  of  the  values  of  {e^h),  which  cor- 
respond to  those  of  (%"')  in  ^^n^\  the  derivatives  of  (1^,)  may  first 
be  taken  with  respect  to  [l^),  and  if  those  of  (^\)  are  afterwards 
taken  with  respect  to  {i]i),  they  give  by  (I8620) 


—  240  — 

Hence,  if  gN*  denotes  the  determinant  of  all  the  quantities  {TFi) 
and  ()/,)  with  reference  to  (^,),  the  equation  (239i3)  gives 

which;  substituted  in  (2392o)  reduces  it  to 

449.  If  there  are  no  equations  of  condition,  the  value  of  ^slUs  is 
reduced  to 

=  (^  +  i),£/),;S D,Lf 

1  71 

1  n 

If  in  this  case,  therefore,  the  values  of  (i]i)  coincide  ivith  those  of  (^i), 
the  multiplier  is  reduced  to  unity. 

450.  If  the  equations  of  motion  were  given  in  the  system  of 
§  310,  in  which  the  forces,  represented  by  the  equations  of  condition, 
are  included  in  those  of  i2,  this  system  might,  by  means  of  the  equa- 
tions of  condition,  be  reduced  to  that  of  Lagrange's  canonical  forms. 
In  performing  this  reduction,  the  equations  of  condition  hold  the 
same  relation  to  the  differential  equations,  which  the  equations  (2IO27) 
hold  to  the  equations  (2IO31),  in  performing  the  reduction  of  §  §  395 
and  425.     It  is  also  obvious  that 

i  i 

Hence  the  divisor  by  which  the  multiplier  of  the  first  of  these 


—  241  — 

systems  is  reduced  to  that  of  the  last,  is  by  (2283),  (222;),  and  the 
preceding  sections 

{:e±d^i]D^ lu A nnD.    nn^    n, d^  //„ f^djp- 

1  n  n  +  1  n  +  2  Zn^        2 

and,  therefore,  the  multipher  of  the  system,  previous  to  reduction, 
is  by  (2408) 

451.  If  the  system  of  differential  equations  is  given  in  Ham- 
ilton's form,  (I663),  the  equation  (21625)  ^^^  the  determination  of  the 
multiplier  becomes 

D log  .dt  +  2:,  {D^  D^-  D^D,;)  11^,^  =  i>log  ^ik  =  0, 

i         i  i         i 

whence  the  multiplier  of  this  system  is  unity. 


CHAPTER    XI. 

MOTION  OF   TRANSLATION. 

452.  If  the  coordinates  of  the  centre  of  gravity  of  a  system 
are  r^-^j^^,  ^g,  and  if  those  of  any  other  point  axa  Xg-{-Xi,y^-\-7/i, 
Zg  -\-  Zi,  the  value  of  T  becomes,  by  (16228)  ^^^  (1642o)  and  the  con- 
ditions of  the  centre  of  gravity  (155i9), 

r=i?^^,m,4-^^,(^;^,e'f). 
31 


—  242  — 

Hence  the  motion  of  the  centre  of  gravity  is  determined  by 
the  equation,  derived  from  (164^2), 

:^,?«,  A.<  =  ^,mMx,  =  D,  n, 

s 

and  the  corresponding  equations  for  the  other  axes.  The  value  of 
£2  may  be  restricted  in  this  equation  to  the  external  forces  and  those 
which  correspond  to  the  external  equations  of  condition,  for  the 
internal  forces  and  equations  of  condition  being  dependent  solely 
upon  the  relative  positions  of  the  bodies  of  the  system,  are  functions 
of  the  differences  of  the  corresponding  coordinates  of  the  bodies, 
from  which  Xg,^g,  ^g  disappear. 

The  motion  of  the  centre  of  gravity  is,  therefore,  independent  of  the 
mutual  connections  of  the  ^mrts  of  the  system,  and  is  the  same  as  if  all  the 
forces  toere  applied  directly  at  this  centre,  provided  they  are  unchanged  in 
amount  and  direction. 

453.  Since  the  second  member  of  (2423)  expresses  the  whole 
amount  of  force,  acting  upon  the  system  and  resolved  in  the  direc- 
tion of  the  axis  of  x,  this  equation  expresses  that  the  motion  of  the  cen- 
tre of  gravity  in  any  direction  depends  upon  the  whole  amount  of  external 
force  acting  in  that  direction. 

If,  therefore,  the  ivhole  amount  of  external  force  acting  in  any  direc- 
tion vanishes,  the  velocity  of  the  centre  of  gravity  in  that  direction  is  uniform. 

MOTION    OF   A    POINT. 

454.  When  the  system  is  reduced  to  a  single  point,  it  becomes 
a  mass  united  at  its  centre  of  gravity,  and  the  only  possible  motion 
is  that  of  translation.  The  position  of  the  point  is  determined  by 
three  coordinates,  which,  combined  with  their  derivatives  and  with 
the  time,  constitute  a  system  of  seven  variables,  and  require,  in  gen- 
eral, six  integrals  for  the  complete  determination  of  the  motion  of 


—  243  — 

the  point.     The  differential  equations  become,  in  this  case,  if  the 
mass  of  the  body  is  assumed  to  be  the  unit  of  mass, 

with  the  corresponding  equations  for  the  other  axes. 

A    POINT    MOVING    UPON    A    FIXED    LINE. 

455.  The  two  equations  by  which  the  line  is  defined  are  two 
equations  of  condition,  which  may  be  denoted  by 

Together  with  their  derivatives,  they  take  the  place  of  four  of 
the  integrals  of  §  454.  0/  the  iivo  remaining  integrals,  when  il  does  not 
involve  the  time,  both  can  le  determined  hy  quadratures  hy  §  445. 

One  of  these  integrals  is,  indeed,  the  equation  of  living  forces 
(IGSig),  which  becomes  in  this  case 

The  final  integral  is  obtained  from  this  integral  by  the  equation 

Dr,S 


456.  It  follows  from  (243i9)  that  the  velocity  of  a  body  only 
depends  upon  its  initial  velocity  and  the  value  of  the  potential  at 
each  point  of  its  path ;  and  this  conclusion  coincides  with  the  propo- 
sition of  §  58.  In  zvJiatever  path,  therefore,  a  body  moves  from  one  j^oint 
to  another,  the  increase  or  decrease  of  the  square  of  its  velocity  may  he  meas- 


—  244  — 

lived  hy  that  of  the  potential,  ivhen  the  equations  of  condition  and  the  forces 
tvhich  act  upon  the  system  are,  like  the  fixed  forces  of  nature,  independent  of 
the  time  and  the  velocity  of  the  hody, 

457.  If  there  is  any  point  upon  the  line,  beyond  which  the 
decrease  of  the  potential  exceeds  one  half  of  the  square  of  the 
initial  velocity,  the  body  cannot  proceed  beyond  that  point.  If  there 
is,  in  each  direction  from  the  initial  position  of  the  hody  upon  the  line,  a  lim- 
iting point  of  this  description,  the  motion  of  the  hody  is  restricted  to  the  inter- 
vening space.  Since  the  body  can  only  have  the  direction  of  its 
motion  reversed  at  the  limiting  points  where  its  velocity  vanishes,  it 
must  oscillate  back  and  forth  upon  the  whole  of  the  intervening 
portion  of  the  line,  according  to  the  law  expressed  by  the  equation 
(24323). 

It  is  evident  from  the  inspection  of  the  equation  (24323),  that 
the  time  which  the  body  occupies  in  passing  from  any  point  (J.)  of 
the  line  to  another  point  (^),  must  be  the  same  with  that  which  it 
occupied  in  the  preceding  oscillation  in  the  reverse  transit  from  the 
point  {B)  to  the  point  (^) ;  and,  therefore,  the  entire  duration  of  oscil- 
lation must  he  invariahle. 

458.  If  the  line  returns  into  itself,  and  if  there  is  no  point 
upon  it  for  which  the  decrease  of  the  potential  is  as  great  as  the 
initial  power  of  the  body,  the  hody  will  continue  to  move  through  the 
ivhole  circuit  of  the  line,  and  ivill  always  return  to  the  same  point  ivith  the 
same  velocity,  so  that  the  period  of  the  circuit  ivill  he  constant. 

459.  When  the  forces  and  the  equations  of  condition  involve 
the  time,  the  multiplier  becomes  by  (238ig) 

and  the  last  of  the  integrals,  which  are  required  to  solve  the  prohlem,  can  he 
ohtained  by  quadratures. 


245  — 


THE    MOTION    OF    A    BODY    UPON    A     LINK,    WHEN     THERE     IS     NO     EXTERNAL     FORCE. 

CENTRIFUGAL    FORCE. 

460.  When  the  line  is  fixed,  and  there  is  no  external  force,  SI  van- 
ishes in  (243i9),  and  the  velocity  is,  therefore,  constant. 

461.  In  this  case,  the  line  may  be  regarded  as  the  locus  of  a 
resisting  force,  which  acts  perpendicularly  to  the  line.  The  plane  of 
X  and  y  may  be  supposed  to  be,  for  each  instant,  that  of  the  curva- 
ture of  the  line  at  the  position  of  the  body,  R  may  be  the  resisting 
force  of  the  line,  and  (>  its  radius  of  curvature  ;  and  elementary  con- 
siderations, combined  with  the  equation  (I6425),  give 

DtX  =  Z>,5  sin  ^  =  z;  sin  ^, 

D^X  :=  V  cos^'Z>^^  =  v^  cos'^D/^  =  -  cos^  =  ii  cos'^, 


whence 


Q 


so  that  the  j^^^ssnre  against  the  line  is  measured  hy  the  quotient  of  the 
square  of  the  velocity  divided  hy  the  radius  of  curvature,  which  is  called  the 
centrifugcd  force  of  the  body. 

462.  If  there  are  external  forces,  the  whole  pressure  iqjon  the  line 
is  ohtained  hy  comhining  the  action  of  all  the  external  forces  resolved  perpen- 
dicidarly  to  the  line,  with  the  centrifugal  force. 

463.  The  centrifugal  force  cannot  be  used  as  a  motive  power 
in  machinery,  for  the  body  moves  perpendicularly  to  the  direc- 
tion of  this  force ;  and,  therefore,  the  power  communicated  by  it 
vanishes,  because  it  is  measured  by  the  product  of  the  intensity  of 
the  force  multiplied  by  the  space  through  which  it  acts. 

464.  If  the  line  is  not  fixed  in  position,  but  has  a  motion  of 


—  246  — 

translation,  the  same  motion  of  translation  may  be  attributed  to  the 
axes  of  coordinates,  so  that  the  coordinates  of  the  moving  origin  at 
any  time  may  be  a^.,  Uy,  a,,  with  reference  to  the  fixed  axes.  If  the 
coordinates  of  the  body  with  reference  to  the  moving  axes  are  b^?  ^</> 
l^,  the  value  of  2  ^T  (1642i)  becomes 

2r=^,(,^;+«;)^ 

=r  DtS'^  -\-2wI)iS  COS  w-\-^y 
2 

^=s'  -{-22vs' COS ,; -|- iv"^ 

if  2v  denotes  the  velocity  of  the  motion  of  the  origin,  and  5  the 
length  of  the  line  passed  over  by  the  body.  Hence  Lagrange's 
equation  (164i2)  gives 

Dt  (/  -f-  tv  cos  /„)  =  Ds  (/  zv  cos ;',)  ==  /  tvD^  cos  ^ . 

But,  since  the  angles  which  5  makes  with  the  axes  are  inde- 
pendent of  the  time,  the  derivative  is 

A  («f  cos  ^,)  =  D,:E^  (tv^  cos  ■;)  =  2^  (?<  cos  ^  +  / tv^  D,  cos  i) 
=  Z^  {w^  cos  %)  -\-  /  D,  {vj  cos ,;) , 

which  reduces  the  preceding  equation  to 

J),s  =  —  2:, {iv^  cos ^)  =  —  TFcos  ^y, 


if 


2  2  2 


denotes  the  acceleration  of  the  line  at  each  instant.  Hence  it  is 
easy  to  see  that  if  the  acceleration  is  perpendicular  to  the  line,  the  relative 
velocity  of  the  hod//  to  the  line  is  not  clmnged  ;  hut  if  the  acceleration  is  in 
the  direction  of  the  line,  the  change  of  relative  velocity  is  exactly  equal  to  the 


—  247  — 

acceleration,  so  that  there  is,  in  this  case,  no  change  in  the  actual  velocity  of 
the  hody  in  space. 

465.  It  follows,  from  the  preceding  investigation,  that  if  the 
motion  of  the  line  is  uniform,  the  relative  velocity  of  the  body  and  the  line 
remains  constant. 

466.  It  is  also  apparent  from  this  investigation  that  even  under 
the  action  of  external  forces,  the  relative  motion  of  the  body  to  the  line  may 
he  computed,  hy  regarding  the  acceleration  of  the  line  as  a  force  acting  upon 
the  body  in  a  direction  opposite  to  its  actual  direction. 

467.  If  the  line  rotates  about  a  fixed  axis,  which  is  assumed  to 
be  the  axis  of  z,  let 

u  be  the  projection  of  the  radius  vector  upon  the  plane  oixy, 
(p  the  angle  which  u,  makes  with  the  rotating  axis  of  x,  and 
a  the  velocity  of  rotation, 

and  the  value  of2T  becomes 

=  (I),sf-{-2u''(f'a-{-u''a'' 

2 

=^s' -\-2uas^  COS ^ -\- li^a^, 

in  which  6  is  the  angle,  which  s  makes  with  the  elementary  arc  udcp. 
Hence  the  derivatives  of  T  are 

D.T^ns  -}-««cos^, 

D, T=  s' D,{ua  cos  &)  +  a^u  cos  ;' ; 

and  the  equation  (164i2)  becomes 

D^,s^=  —  u  cos  t1  a  -j-  a'^u  cos  ". 

The  former  of  the  two  terms  which  compose  the  second  mem- 


—  248  — 

ber  of  this  equation,  is  the  negative  of  the  acceleration  of  the  rotar 
tive  velocity  resolved  in  the  direction  of  the  arc  of  the  rotating  line. 
The  latter  term  represents  the  centrifugal  force,  which  corresponds 
at  the  body  to  the  rotation  {a),  and  which  is  also  resolved  in  the 
direction  of  the  moving  arc.  But  the  centrifugal  force  is  purely 
relative  in  its  character,  and  arises  from  the  resistance  of  the  body 
to  accompany  the  curve  in  its  change  of  motion  occasioned  by  rota- 
tion. These  terms  combined  show,  then,  that  in  this  case,  as  well  as 
in  that  of  translation,  and,  consequently,  in  every  case  the  relative 
motion  of  the  hod//  to  the  line  may  he  ohtained  hy  attributing  to  the  hody  the 
negative  of  the  acceleration  of  the  line,  ivhich  occurs  at  the  position  of  the 
hody ;  in  the  case  of  external  forces,  their  action  must  he  united  to  that 
ivhich  arises  from  the  acceleration  of  the  line. 

468.  In  the  case  of  an  uniform  rotation  about  a  fixed  axis,  the 
equation  (24729)  becomes 

Z^  5  =  «^  II  cos  "  =  a^  u  D,  u . 

The  integral  of  the  product  of  this  equation,  multiplied  by 
2^5,  is 

{D,sf^a\u'  +  A), 
in  which  A  is  an  arbitrary  constant.     Hence  it  is  obvious  that 

469.  When  the  constant  [A)  is  negative,  the  value  of  u  cannot 
be  less  than  y/  —  A;  so  that  when  the  body  approaches  the  axis,  its 
velocity  upon  the  line  is  constantly  retarded,  and  vanishes,  when  its 
distance  from  the  axis  is  reduced  to  y/ — A,  after  which  the  direction  of 
the  motion  is  reversed.  If  the  portion  of  the  line,  upon  which  the 
body  moves,  extends  at  each  extremity,  so  as  to  be  at  as  small  a  dis- 


—  249  — 

tance  as  y/  —  A  from  the  axis,  the  hody  iiill  oscillate  upon  it  with  a  con- 
stant jyeriod  of  oscilkdion. 

470.  When  the  constant  [A)  is  positive,  or  when  it  is  negative, 
and  no  joortion  of  the  line  in  the  direction,  towards  which  the  body 
is  moving,  is  at  so  small  a  distance  as  y/  —  A  from  the  axis,  the 
motion  of  the  body  upon  the  line  will  constantly  retain  the  same 
direction.  If,  moreover,  the  curve  returns  into  itself,  the  hody  ivill 
always  continue  to  move  around  it,  with  a  constant  period  of  revolution. 

471.  When  the  constant  (^)  vanishes,  the  equation  (2482o) 
gives 

DiSr=.uU 

Js  U  J  a        " 

If  the  curve,  also,  passes  through  the  axis  of  rotation,  the  value 
of  i>„5  may  be  supposed  to  be  constant,  while  the  body  is  very  near 
the  axis,  and  may  be  represented  by  (i ;  so  that  the  motion  of  the 
body  in  the  vicinity  of  the  axis  is  given  by  the  equation 

a  / 1=  ^j  log  u . 

The  second  member  of  this  equation  becomes  infinite  when  u 
vanishes,  and,  therefore,  the  motion  of  the  hody,  in  this  case,  is  infinitely 
sloiv  in  the  immediate  vicinity  of  the  axis. 

472.  When  the  rotating  line  is  straiyM,  let 

p  be  the  distance  of  its  nearest  approach  to  the  axis  of  rotation,  and 
^  the  angle  which  it  makes  with  the  plane  of  xy. 

.  If  then  s  is  counted  from  the  foot  of  the  perpendicular,  which 
joins  the  nearest  points  of  the  line  and  the  axis  of  revolution,  the 

32 


—  250  — 
value  of  i^  is  given  by  the  equation 

ll"  Z=L  f  ■\- i  (IQ^- ^  '^ 

whence  (24804)  becomes,  in  this  case, 

=  ^-l^log[.cos^  +  V(/+^+.^cos^^)]-^^^|f^; 

in  which  the  arbitrary  constant  is  determined  so  that  t  may  vanish 
with  5,  and  this  equation  is  apphcable  when  (jo^  -)-  A)  is  positive. 
In  this  case,  the  substitution  of  the  notation 

,                 h 
tan  9== -, 

'  S  COS  d 

reduces  the  preceding  equation  to 

a  t  cos  ^  =  log  cot  h  9 . 

But  when  {^f--\-A)  is  negative,  the  substitution  of  the  notation 

Ti'^  =  -{f-\-A), 
h 


smif 


5  cos  5 


and  the  determination  of  the  arbitrary  constant,  so  that  t  may  vanish 
when  s  has  its  least  possible  value  of  Jc  sec  ^,  reduce  the  equation 
(2506)  to 

a  t  cos  ^  =  log  tan  ^  1// . 


—  251  — 

When  (//  -|-  A)  vanishes,  the  equation  (250o)  is  reduced  to 

e 

a  t  cos  6  =  log  — ; 

So 

in  which  Sq  is  the  initial  value  of  s.  When  p  also  vanishes,  the  sur- 
face described  by  the  line  is  a  right  cone,  and  when  it  is  developed 
into  a  plane,  the  path,  described  hy  the  hodi/,  becomes  a  bgarithmic  spiral. 

473.     When  the  rotating  line  is  the  circumference  of  a  circle  ivhich 
is  situated  in  the  plane  of  rotation,  let 

R  denote  the  radius  of  the  circle, 

a  the  distance  of  the  centre  of  the  circle  from  the  origin, 

2  ip  the  angle,  which  the  radius  of  the  circle,  drawn  to  the  body, 

makes  with  that  which  is  drawn  in  a  direction  opposite  to 

the  origin, 

and  the  equation  (24824)  becomes 

2R 


at 


=/ 


^lA-\-{R-\-af—4.aEs.m^(f'\ 


When  A-\-(R  —  a)^  h positive,  which  corresponds  to  the  case  of 
§  470,  let 

h^=.A  +  (R  +  a)\ 

AaR 


BlV^i 


W 


and  by  the  notation  of  elliptic  integrals  of  §  169,  the  equation  (251i8) 
becomes 

2  R  rr 

When  i  is  so  small  that  its  fourth  power  may  be  rejected,  this 


—  252  — 
equation  gives,  by  an  easy  reduction, 

«zf  =  (1  -[-  4  sin^2)— y-^  —  jT  sin^2;sin2g5. 

In  this  case,  therefore,  the  time  of  describing  the  semicircum- 
ference,  for  which  2  cp  is  greater  than  a  quadrant,  exceeds  the  time 
of  describing  that  for  which  2(p  is  less  than  a  quadrant  by 

T-siirt 


h  K'    ~\_A-\-{R-\-  afj ' 

When  A -\- {li  —  «)^  is  nc(/at ive,  \v\iich.  corresponds  to  the  case 
of  §469,  let 

surt=z- — -, 

AaJi 


sin  (5 


sin(p 


and  the  equation  (251i8)  becomes 

2R    r         ,         2Rsmi    C 
at-^^-  I  sec  6  =  — -. —   /  sec 9) 


When  i  is  so  small  that  its  square  may  be  rejected,  the  duration 
of  an  oscillation  becomes 

ay    a 

When  the  circumference  passes  through  the  axis  of  rotation,  a 
is  equal  to  7?,  and  the  time  of  the  small  oscillation  becomes  identical 
with  that  of  the  semi-revolution  of  the  circle;  but  the  time  of  a 
larger  oscillation  exceeds  that  of  the  semi-revolution. 


—  253  — 
When  A-\-  (R  —  a)^  vankhcs,  the  equation  (251i8)  becomes 

When  A-[-{R  —  ciY  is  very  small,  and  its  ratio  to  ^ciR  is 
denoted  by  d li,  the  equation  (251i8)  gives  throughout  the  greater 
portion  of  the  path,  in  which  (f  difiers  sensibly  from  ^  n,  that  is,  in 
which  the  body  is  not  near  its  point  of  closest  approach  to  the  axis 
of  rotation,  so  that  the  square  o^  ^  A  may  be  neglected, 

=  (1  —  ^d  A)  i/— log  tan  (j  ti  -\-  i(p)  —  }dAi/~t(in(psec(p. 

But  in  the  vicinity  of  the  point  of  nearest  approach,  let 

If  =  in  —  (p 

be  so  small  that  its  square  is  of  the  same  order  with  d  A,  and  the 
equation  (251i8)  gives 

at=z  —  v^-  I     .-,,...  =  —  v^ -  Sin^~^^  -=,  when  d A  is  positive. 
=  — i/—  Cos^~-^^  ,  -  '      .,,  when  dAis  neo;ative, 

474.  When  the  rotatinc/  line  is  ivholly  contained  upon  the  surface  of  a 
cylinder  of  revolution  of  zvhich  the  axis  is  the  axis  of  revolution,  u  is  con- 
stant and  the  equation  (24729)  becomes 

D.sl)^s=^  —  u^a  , 

from  which  (p  or  s  may  be  eliminated  by  the  given  equation  of  the 
curve. 


—  254  — 

475.  When  the  velocity  of  rotation  is  constant,  the  second 
member  of  (25328)  vanishes,  and  the  velocity  of  the  body  is  conse- 
quently uniform. 

476.  When  the  curve  is  a  M/o:,  the  value  ofi).s  is  constant, 
and  the  equation  (25829)  gives 

in  which  A  is  an  arbitrary  constant. 

477.  When  the  acceleration  is  uniform,  a  is  constant,  and  the 
integral  of  (25328)  gives 

iij^a't^z    i  --— ; 

in  which  A  is  an  arbitrary  constant. 


MOTION    OF   A   HEAVY   BODY    UPON    A   FIXED    LINE.      THE    SIMPLE    PENDULUM. 

478.  When  the  line  is  fixed,  and  the  force  which  acts  upon 
the  body  is  that  of  gravity  at  the  surflice  of  the  earth,  represented 
by  ff,  and  the  axis  of  2  is  assumed  to  be  the  vertical,  directed  down- 
wards, the  equations  (243i9_32)  give 

t-c 1 r     A^ 

479.  If  the  curve  is  contained  upon  the  surface  of  a  cylinder 
of  which  the  axis  is  vertical,  the  motion  of  the  body  is  the  same  as 


—  255  — 

it  would  be  upon  the  plane  curve,  obtained  by  the  development  of 
the  cylinder  into  a  vertical  plane ;  because  the  value  of  D^s  is  not 
changed  by  this  development. 

480.  If  the  fixed  line  is  straight,  the  equation  (2542s)  becomes 

cos  i  gt  =  \l{2gz  J^2H)  —  sl{2H)  =  v  —  v,, 

if  Vq  is  the  initial  velocity  of  the  body. 

481.  If  there  is  no  initial  velocity,  the  preceding  equations 
become 

(ioslgt=zs^{2gsQO^l)^v, 
or 

^  V  2  s  v^  z 

482.  If  the  curve  is  the  circumference  of  a  circle,  the  centre 
of  the  circle  may  be  assumed  as  the  origin  of  coordinates.  If  then 
the  axis  of  %  is  the  intersection  of  the  plane  of  the  circle  with 
the  vertical  plane,  which  is  drawn  perpendicular  to  it  through  the 
origin,  and  if  E  is  the  radius  of  the  circle,  and 

2y  =  (y, 

the  equation  (25428)  becomes 

2E  r  27? 


r 2ji r 

—  J.sJ{2gIicoslcos2(p^2H)  ~  J. 


^\J{2gIicoslcos2(p-\-2JI)        J  ^sj  {2  H  -{-2  g  R  co&l  —  ^gRcosl  sin^  g))  * 


If  then  II  is  greater  than  g  Bcos\,  which  is  similar  to  the  case 
of  §  470,  let 

h^=2ff+2gBcosl^, 

ig  R  cos  1=^1^  sin^« ; 


—  256  — 
and  tlic  preceding  equation  becomes 

AVhen  i  is  quite  small,  this  equation  admits  the  same  reduction 
with  that  given  (25l3i — 2523). 

If//  is  smaller  than  ^i?  cos  ^i,  which  is  similar  to  the  case  of  §469, 
let 

/^2  z=.  4:(/Bcosl  sin^i 

.     .         sin  qp 
sin  I 

and  the  equation  (25624),  becomes,  by  the  same  reduction  with  that 
given  in  (2521^), 


V  Vycosj^/ 


which  when  i  is  small  gives  for  the  time  of  oscillation  of  the  simple  pen- 
dulum in  an  oblique  plane 

V  5'cos^j 
If//  is  just  equal  to^7?cos^^,  the  equation  (25504)  becomes 
^  =  V^(7^)l°Stan(J.  +  ic;). 

The  case  in  which  H  differs  but  little  from  ^7?  cos?,,  may  be 
subjected  to  the  same  treatment  with  that  adopted  in  (2555_23). 


—  257  — 


MOTION    OF    A    HEAVY    BODY    UPON    A    SIOVIN'G    LINE. 

483.  If  the  heavy  body  moves  upon  a  line,  which  has  a 
motion  of  translation  in  space,  the  equation  of  motion  becomes,  by 
the  form  of  argument  and  notation  adopted  in  §  464, 

Z>^s=:  —  TFcos,', -l-^cos*. 

484.  If  the  motion  of  the  line  is  uniformly  accelerated  and 
invariable  in  direction,  the  motion  of  the  body  upon  the  line  is  the 
same  which  it  would  be  if  the  line  were  fixed,  and  the  force  a  con- 
stant force  which  coincided  in  amount  and  direction  with  the  re- 
sultant of  ^  and  —  W.  Thus  if  the  line  moves  vertically  downwards 
with  an  accelerated  velocity,  equal  to  that  of  a  heavy  falling  body, 
the  body  moves  upon  the  line  with  an  uniform  velocity. 

485.  If  the  line  is  straight,  and  if  the  motion  of  translation 
follows  a  law,  dependent  exclusively  upon  the  time,  so  that  if 

At  denotes  the  law,  by  which  the  line  moves  in  the  direction 
of  its  length,  the  acceleration  in  the  direction  of  the  line  is 

—  TFcos4.=  i>r  A; 
and  the  value  of  s  becomes 

in  which  a  and  h  are  arbitrary  constants.  The  absolute  motion  of 
the  point  in  any  direction  in  space,  as  that  of  the  axis  of  x^ ,  is  repre- 
sented by  the  equation 

.Ti  =  (S  —  A)  cos  ^,^p  COS  l^, 

33 


—  258  — 

in  which  j9  denotes  the  perpendicular  upon  the  line  from  the  origin. 
If  the  line  is  vertical,  and  limited  in  its  motion  to  the  vertical  plane 
of  Xi ;?!,  and  if  the  axis  of  %  is  vertical,  the  equations  which  deter- 
mine the  position  of  the  point  in  space  are 

When  j)  increases  uniformly  so  that  p  is  constant,  these  equa- 
tions give 


Xi  =:zp  t, 


Zi  —  a  -\-  —,  Xi  -\-  — V2  ■^i  > 


so  that  the  path  of  the  body  in  Space  is  a  parabola,  of  which  the 
axis  is  vertical. 

486.     If  the  line  moves  ivith   an  uniform  motion  in  a  straight  line, 
the  equation  (2578)  gives 

I>^tS  =  g  cos  l. 
The  integral  of  the  product  if  this  equation  multiplied  by  2  D^s  is 

{D,sf=f^2gcosiD,s  =  2gJ^D,z=^2gz  +  a, 

in  which  a  is  an  arbitrary  constant.     Hence  if 

V  denotes  the  velocity  of  the  translation  of  the  line, 

the  square  of  the  velocity  of  the  point  in  space  is 

(i>,.,)2=[v/(2^^  +  «)-^cosrp  +  (Fsinrf 

^2gz-\-a-{-V'-  —  2Vcos]\l{2gz+a). 

The  augmentation  of  the  power  of  the  moving  body  above  its 


—  259  — 
initial  power  is,  then, 

If  the  body  iiad  moved  through  the  same  path  upon  a  fixed 
curve,  the  increase  of  power  would  have  been 

^=^(.--^J+^F^fcosr. 

If  P  is  greater  than  Q,  the  excess  of  P  above  Q  is  the  power 
acquired  by  the  body  from  the  accelerating  motion  of  the  line.  But 
if  Q  exceeds  P,  the  excess  of  Q  above  P  is  the  power  communi- 
cated by  the  body  to  the  line,  which  involves  the  theory  of  many 
machines,  of  which  heavy  bodies  are  the  moving  forces.  If,  for  ex- 
ample, the  line  moves  horizontally,  the  power  communicated  by  the 
weight  is 

Q  —  P  =  V(v  cos  [—  v°  cos  Ic). 

If,  moreover,  the  initial  velocity  of  the  bod}^,  relatively  to  the 
line,  vanishes,  the  expression  of  the  communicated  power  is  re- 
duced to 

and  when  the  direction  of  the  line  at  its  extremity  coincides  with 
that  of  its  translation,  this  expression  is  still  further  reduced  to 

487.  If  the  line  is  the  circumference  of  a  vertical  circle,  of  which 
the  radius  is  P,  and  if  9  is  the  angular  distance  of  the  body  from 
the  lowest  point  of  the  circumference,  the  equation  of  motion  (2578) 
becomes 

R]y\{^z=L — TFcos^ — ^sin^. 


—  2G0  — 

When  the  motion  of  the  line  is   in  a  vertical   direction  this 
equation  becomes 

BI>}(f  =  —  {  W-\-ff)  sin  (f  ; 
which,  when  (f  is  very  small,  is  reduced  to 

The  integral  of  this  equation  is 

in  which  A  and  b  may  be  determined  by  the  equations 

2sJ{ffB)  DAogA  =  -  TFsin  2  (^ |r+  b), 

2^{ffB)I),b  =  WA[l—cos2{isJ§  +  b)]', 

which  give 

s/{ffB)D,{Asmb)  =  —  AWsm(tJ^  +  b)sin(tJQ 

=  ^Aw[cos(2tJ^+b)  —  cosb] 

=^Aw[sm''ib—siii'(tJj^-\-H]; 

s/{c^B)I),{Acosb)^  —  AWsm(t^^+b)cos{t^^) 

=  —iAw[Hm(2i^l^+b)  +  smb]; 

when  W  is  very  small  in  comparison  with  y,  A  and  B  may  be  as- 
sumed to  be  constant  in  the  first  integration  of  the  second  members 
of  these  equations. 

When  W  is  dependent  upon  the  position  of  the  body  in  such 


—  261  — 
a  way,  that,  if  '^  is  a  function  of  time, 

W(p  =  A  TFsin  (^  y/j  -{-b)  =  ^; 
the  preceding  equations  give 

V/(yi?)^sin5  =  — J(ersin(/y/|)), 
s/{ffIi)Acosh  =  -f{^cos{tsJg). 

If,  for  example, 

'tf=  2hsm?nt ; 
these  integrals  become 

"•     m- R — g 

in  which  the  arbitrary  constants  are  determined  so  that  A  and  h 
vanish  with  the  time. 

488.  If  the  line  rotates  about  the  vertical  axis  of  z,  the  equation 
of  motion  becomes,  by  the  analysis  and  notation  of  §  467, 

jyfs  =  —  iicos^a' -\-a^u cos" -\-(/ cos i 
=  —  wcos^a'  -\-a^uDsU-\-(/  D,z. 

489.  When   the   rotation   about   the   vertical  axis  is   uniform,  this 
equation  becomes 

The   integral  of  the  product  of  this  equation   multiplied  by 


—  262  — 
2D,s  is 

in  which  a  is  an  arbitrary  constant. 

490.  When  the  rotating  line  is  straight  and  passes  at  a  distance 
p  from  the  axis,  if  s  is  counted  from  the  foot  of  the  perpendicidar 
{p)  '^V^^  ^^^^  VmQ,  the  equation  becomes 

(Btsy  =:  a^s^  sin^*  -|-  2^s  cos^  -f"  ^V  +  ^ 

r=  (a  s  sin  ^  +  ^  cot '.)+«  +  «y  —  (^  cot :)  , 

of  which  the  integral  is  easily  found  to  be 

a^sin  *  =  log(a^5sin^^4"  2^cos^  +  2a  sin  *Z>,5)  +  3, 

in  which  h  is  an  arbitrary  constant. 

491.  The  integral,  in  this  case,  can  be  just  as  readily  obtained 
from  the  equation  (26I29)  which  becomes  a  linear  differential  equa- 
tion.    Its  direct  integral  is 

or  cos  2  .     ats'mt    ,     -j-,    — atsmf. 

«''siir|  ' 

in  which  A  and  B  are  arbitrary  constants.  This  form  is  identical 
with  that  given  by  Vieille  in  his  solution  of  the  particular  case 
of  this  problem,  in  which  p  vanishes. 

492.  If  «<(^^cot:y— aV 

the  value  of  s  must  be  such  as  to  render  the  second  member  of 
(2629)  positive ;  that  is,  the  limiting  values,  between  which  the 
body  cannot  be  contained,  are  defined  by  the  equation 

«5osin^  =  — ^cot^+i/[(^cot^)'  — «V  — «]. 


—  263  — 

The  velocity  of  the  body  upon  the  line  vanishes  at  these  limits. 
If  the  initial  direction  of  the  motion  of  the  body  is  towards  these 
limits,  it  will  approach  them  with  a  diminishing  velocity ;  and 
when  it  arrives  at  the  nearest  limit,  the  direction  of  motion  w^ill  be 
reversed,  and  it  will  thenceforth  continue  to  move  away  from  the 
limits. 

If  «  =  —  a- J? 

one  of  the  limits  is  at  the  foot  of  the  perpendicular  (/>),  and  the 
other  limit  is  above  this  foot,  at  the  point  for  which 

5  = ^cot" 

If  «  <C  —  ay, 

one  of  the  limits  is  above  the  foot  of  the  perpendicular,  while  the 
other  is  below  it.     But  if 

while  it  satisfies  the  condition  (26225),  both  the  limits  are  above 
the  foot  of  the  perpendicular. 

493.    If  «>(^cos,'y— «y, 

the  motion  will  always  continue  in  the  same  direction  along  the  line, 
{a  -|-  «y )  will  express  the  square  of  the  velocity  of  the  body  upon 
the  line  when  it  is  at  the  foot  of  the  perpendicular.  The  point  of 
least  velocity  upon  the  line  will  be  determined  by  the  equation 

g  cos  I 


and  the  least  velocity  will  be 


—  264  — 

494.  If  «  =  (^^cot,^)'— «V 

the  direction  of  the  motion  along  the  line  is  not  subject  to  reversal ; 
for,  in  this  case,  the  equation  (2629)  becomes 

DtS  :=.  a  s  sin  I  -\--  cot '^ ; 
of  which  the  integral  is 

((jl  s  Sin  ^  \ 

^  cos  J      '       / 

The  time  of  reaching  the  point,  at  which 

g  cos  I 

a^sin^j' 

that  is,  the  point,  at  which  the  velocity  vanishes,  becomes  infinite ; 
or  in  other  words,  the  body  never  reaches  this  point,  at  which  its 
direction  of  motion  is  to  be  reversed ;  or  if  the  body  is  placed  at  this 
point  without  any  initial  velocity  along  the  line,  it  will  remain  sta- 
tionary upon  the  line. 

495.  If  the  7'otating  line  is  the  circumference  of  a  circle,  of  which 
the  radius  is  R,  let  the  origin  be  assumed  so  that  the  centre  of  the 
circle  may  be  upon  a  level  with  the  foot  of  the  perpendicular  (jo),  let 
fall  from  the  origin  upon  the  plane  of  the  circle.     Let  then 

h  denote  the  distance  of  the  centre  of  the  circle  from  the  foot 
of  the  perpendicular, 

9)  the  angular  distance   upon  the   circumference  of  the  body 
from  the  lowest  point  of  the  circumference, 

and  the  values  of  z  and  ii,  in  equation  (2622),  arc  given   by  the 


—  265  — 

equation 

0r=  li  COS  (f  sin^-|-^cos^, 
u^  z={/c-{-  li  sin  (pf-\-  {p  sin  I  —  7?  cos  (p  cos  ^f 

whence  equation  (2622)  becomes 

-\-2{ff  —  a^p  cos^)  it  sin^  sin  cp  —  «^i?^sin^^  cos^  (p . 

The  points  of  maximum  and  minimum  velocity  along  the  arc 
are,  therefore,  determined  by  the  equation 

a^kM  cos  (pi  —  ((/  —  a^p  cos  ^)  i?sin^  siu^j  -|-  «^i?^sin^^  sin  ^i  cos  9)1  =  0, 

and  are,  consequently,  at  the  intersections  of  the  circumference  with 
the  equilateral  hyperbola,  which  is  described  in  the  plane  and  passes 
through  the  centre  of  the  circle,  of  which  one  of  the  asymptotes  is 
horizontal,  and  the  polar  coordinates  (r2,  (p^)  of  the  centre,  with 
reference  to  the  centre  of  the  circle,  are  given  by  the  equations, 

^2  sin  ^2  =  —  ^'^  cosec^  I , 

^2  cos  ^2  =  ^  cosec^  — 2^  cotf . 

This  hyperbola  cannot  cut  the  circumference  in  less  than  two 
points ;  and  there  are  four  points  of  intersection  when  the  distance 
from  the  centre  of  the  circle  to  the  nearest  point  of  the  branch 
of  the  hyperbola,  which  does  not  pass  through  it,  is  less  than  the 
radius  of  the  circle.  The  polar  coordinates  [r^,  ^3)  of  this  nearest 
point  of  the  second  branch  of  the  hyperbola  are  given  by  the 
equations 


tan  (p3=\J  tan  (p2, 
34 


rg  ■=  r^  cos  (po,  sec^  (^3 . 


—  266  — 

496.  When  the  body  is  originally  placed  at  one  of  the  points 
of  maximum  or  minimum  velocity,  without  any  initial  velocity 
along  the  circle,  it  remains  stationary  upon  the  curve  ;  but  its 
position  upon  the  curve  is  one  of  stable  equilibrium,  when  it  is 
placed  at  a  point  of  maximum  velocity,  and  a  position  of  unstable 
equilibrium,  when  it  is  placed  at  a  point  of  minimum  velocity. 
When  the  body  is  originally  placed  upon  the  curve,  without  any 
initial  velocity  along  the  line,  at  a  point  different  from  these  points 
of  maximum  or  minimum  velocity,  it  oscillates  about  that  point 
of  greatest  velocity  from  which  it  is  not  separated  by  a  point  of 
least  velocity ;  its  oscillations  embrace  both  the  points  of  great- 
est velocity,  when  the  velocity  is  sufficient  to  carry  it  through 
either  of  the  points  of  least  velocity,  that  is,  when  the  velocity, 
which  corresponds  to  the  initial  point  in  the  general  equation 
(265;),  is  less  than  that  which  corresponds  to  one  of  the  points 
of  least  velocity.  When  the  initial  velocity  of  the  body  is  greater 
than  the  excess,  which  is  given  by  equation  (265;)  of  the  velocity 
at  the  initial  point  above  the  least  of  the  minimum  velocities,  the 
body  constantly  moves,  in  the  same  direction,  through  the  entire 
circumference. 

497.  The  case  in  which  the  initial  velocity  of  the  body  is 
just  equal  to  the  excess,  which  is  given  by  equation  (2667)  ^^  the 
velocity  at  the  initial  point  above  either  of  the  minimum  veloc- 
ities, admits  of  integration.  In  this  case,  it  is  easy  to  express  the 
equation  (2667)  in  the  form 

R  [Dt  cj))^  =  2  a^ k  (sin  c/)  —  sin  f/ij)  —  a^R  m.v?l  (cos^  (p  —  cos^  (pi) 
-[-2  ([/  —  a^p  0,0^1)  sin^(coscj)  —  cosg)i), 

which  by  means  of  (2652o)  assumes  the  form 

R{D^(.pf=za^i^iv?l  [2^2  cos  (9  -{-^>2)  —  2 /a  cos  (9^1 -f- 92) 
—  R  cos^  ip  -\-  R  cos^  9)1] . 


—  267  — 

The    condition    for   the    determination    of  the    point   of  min- 
imum velocity  gives  also  the  equation 

2 ^2  sin  {(pi  -\-  (f>2)  =  ^  sin  2  (pi , 

which  substituted  in  the  previous  equation  with  the  notation 

^=^(p  —  (pi) 


H 


sin  ('jTi  — T'2) 


sin  (gii  +  qpjj) 

gives 

{D,  <Pf  =  i  a^  ^ij^2  p  gjj^2  ^  ["cog  2  ( </>  +  (pi)  —  //]  . 

If,  therefore,  H  is  negative  and  absolutely  greater  than  unity, 
that  is,  if  (p^i  is  not  in  the  same  quadrant  with  (p2,  the  value  of 
fp  is  unlimited  ;  but  if  //  is  less  than  unity,  the  limits  of  <f>  are 
given  by  the  equation 

cos2(^4-f/)i)— //. 

The  integral  of  the  equation  (267ii)  is 

at  sml^(icos2(pi—^ II) 

1        sin  (<Z>  +  f/i)  v/(cos2(jri  —  B) — sin  qri  v/Ccos  2  (d) -{- (fi)  —  H'] 

o  cos  (a>  +  <?i)  V^  (cos  2q)i  — ^)  +  cos  g^  ^  [cos  2  ((!)-{- q,^)  —  H^  ' 

498.  If'  the  rotating  line  is  a  iiardbola,  of  ivJiich  the  transverse 
axis  is  vertical,  let 

q  be  the  distance  from  the  vertex  to  the  focus  of  the  parabola, 

and  let  the  origin  of  coordinates  be  assumed  to  be   upon  a  level 
with  the  vertex,  and  let 

Jc  denote  the  distance  of  the  vertex  from  the  foot  of  the  perpendicu- 
lar {p)  let  fall  from  the  origin  upon  the  plane  of  the  parabola. 


—  268  — 

If  the  axis  of  x^  is  the  horizontal  line,  which  is  drawn  in  the 
plane  of  the  parabola  through  its  vertex,  and  if  the  vertex  is 
the  origin  of  .^•^,  the  values  of  s  and  u  are  given  by  the  equations 

and  the  equation  (2622)  is  reduced  to 

(A  sf  =  a^f  +  a-{k-\-  x,)^  +  '-^  +  a 


=  (^-+i)(^.^0^ 


The  integral  of  this  equation,  in  its  general  form,  can  be 
obtained  by  elliptic  functions.  The  point  of  least  velocity  along 
the  curve  is  determined  by  the  equation 

but  there  is  no  such  point,  when 

When  this  latter  condition  is  satisfied,  and  also 

the  velocity  of  the  body  along  the  curve  is  constant. 
When  k  vanishes  and 

the  equation  (268]o)  becomes 


—  269  — 

so  that,  in  this  case,  the  horizontal  velocity  of  the  body  upon  the 
plane  of  the  parabola  is  constant. 

499.     In  the  especial  case,  in  which  the  initial  velocity  is  that 
which  corresponds  to  the  vanishing  of  the  minimum  velocity,  let 

Xi  be  the  value  of  x^  for  this  point  of  minimum  velocity, 

and  the  integral  of  the  equation  of  motion  is 

2  ?i!  \/  («'  +  fj)  =  V/  (-^5  +  4  J^)  +  X,  log  \_x,  +  \/  (.f5  +  4  f)\ 


X^  Cl'2 


500.  ^V^len  the  axis  (k)  of  rotation  is  not  vertical,  the  equation 
of  motion  is  still  reduced  to  the  form  (26I24),  and  tvhen  the  rotation 
is  uniform,  it  becomes 

501.  Wlien  the  rotating  line  ahoiit  the  inelined  axis  is  straight,  if 
the  point  of  the  axis  of  rotation  which  is  nearest  to  the  rotating 
line  is  assumed  as  the  origin,  let 

p  be  the  perpendicular  upon  the  line  from  the  origin, 

let  s  be  counted  from  the  foot  of  the  perpendicular  {p),  and  the 
time  from  the  instant,  when  the  plane  of  the  directions  of  the  axis 
and  the  rotating  line  is  vertical.  The  values  of  u  and  cos ,  are 
given  by  the  equations 

x\^-==.f'  -|-5^sin^'', 
cos^  =  cos ^  cos  ^-f-  sin^  sin^  cos  (a  t) ; 


—  270  — 
which  reduce  the  equation  (269i8)  for  this  case  to 

Dfs=z  a^s  sin^^  -f-y  cos )  cos  I  -\-gdn  ]  sin  ^  cos  («  t) . 
The  integral  of  this  equation  is 
s  =  Ac  '4-Bc  '—^   ,  :  .„  ' /,,     '  .  .:,cos(at), 

in  which  A  and  i?  are  arbitrary  constants. 

502.  If  in  the  general  case  of  the  rotation  of  a  i^lane  curve  about  the 
inclined  axis  the  time  is  computed  from  the  instant,  when  the  plane 
of  the  curve  is  vertical  the  expression  of  (^)  is  given  by  the 
formula 

cos  I  =  cos^  cos  ^  -|-  sin  ]  sin  ^  cos  at. 


MOTION    OF   A    BODY    UPON   A    LINE    IN     OPPOSITION    TO     FRICTION,    OR    THROUGH    A 

RESISTING    MEDIUM, 

503.  The  forces  of  nature,  which  resist  the  motions  of  bodies, 
are  of  various  kinds  and  subject  to  different  laws.  While  their 
philosophical  discussion  must  be  reserved  to  its  appropriate  place, 
it  is  sufficient  for  the  present  purpose,  to  recognize  them  as  forces, 
which  are  opposed  to  the  motion  of  bodies,  and  which  depend  in 
general  upon  the  relative  motions  of  the  body  and  of  the  origin 
of  the  resistance,  whether  this  origin  be  solid  or  fluid. 

504.  If  either  of  the  resisting  forces  is  denoted  by  Si,  and 
if  {I )  denotes  the  angle  which  the  direction  of  its  action  makes 
with  the  path  of  the  body,  the  resistance  to  the  motion  of  the 
body  in  its  path  will  be  expressed  by  /Si  cos '  ,  which  may  be 
immediately  introduced  into  the  equation  of  motion. 


—  271  — 

505.  If  the  hody  moves  upon  a  fixed  line,  the  equation  of 
motion  (243i9)   becomes 

If  there  is,  likeivise,  no  motion  in  the  resisting  medium,  all  the 
forces  of  resistance  can  be  combined  in  one,  which  is  directly 
opposed  to  the  motion  of  the  bodj,  and  the  preceding  equation 
assumes  the  form 

D,s'  =  D,n  —  S. 

506.  If  there  is  no  external  force,  these  equations  become 

A/=^i(>S'iC0sy, 
D,s  =^  —  S. 

507.  The  integral  of  the  latter  of  these  equations  is 

Let  S  have  the  form 

S=a-\-hs'  -\-es'\ 
hi  which  a  and  e  are  positive,  in  the  case  of  nature,  and 

h  +  \l{iae)>0, 

because  S  is  always  positive  when  /  is  positive.     The  correspond- 
ing integral  of  (27I17)  is 

in  which  A  is  an  arbitrary  constant,  and   the  former  integral  cor- 


—  272  — 

responds  to  the  case  of  P<^Aae,whi\e  the  latter  corresponds  to 
b^^Aae.  The  velocity  vanishes  after  the  time  ^o  given  by  the 
equation 

4  =  ^-.;..    '_.-.tan^-^^; 


These  values  are  infinite  in  form,  when 

P  =:z  Aae; 
but,  in  this  case,  the  integral  is 

, A  _\  ^^  /d_J_         ^ 

^  —  ^  "I    b{bs'-\-2a)  —  ^    I    2 77+ * ' 

so  that  the  velocity  vanishes,  when 


\J{ae)' 

These  values  become  infinite  in  form  when  both  h  and  e 
vanish,  but,  in  this  case,  which  includes  that  of  friction  upon  a 
straight  path,  the  integral  is 

a  a 

and  the  instant,  at  which  the  velocity  vanishes  is  determined   by 
the  equation 

When  a  vanishes,  the  value  of  t(^  is  actually  infinite,  so  that 
the  velocity  of  the  body  can  never  be  wholly  destroyed  by  any 
such  form  of  resistance.     It  would  seem,  from  the  preceding  equa- 


—  273  — 

tions,  that  the  direction  of  motion  would  be  reversed  after  the 
time  (4).  But  this  conclusion,  which  is  absurd,  because  it  would 
give  a  resistance  the  power  of  creating  motion,  arises  from  the 
defective  forms  of  notation  w^hich  do  not  express  the  solution  of 
continuity  corresponding  to  the  abrupt  ceasing  of  the  friction  at 
the  instant  of  the  suspension  of  motion. 

508.  When  the  resistance  is  simply  that  of  friction  arising  from 
the  pressure  of  tJie  moving  body  upon  the  line,  to  ivhich  its  motion  is 
restricted,  let 

p  denote  the  direction  of  the  perpendicular  to  the  fixed  line, 
which  is  drawn  in  the  common  plane  of  the  direction  of 
the  external  force  and  of  that  of  the  line, 

dv  the  elementary  angle  made  by  two  successive  radii  of  cur- 
vature to  the  fixed  line,  and 

a  the  coefficient  of  friction, 

and  the  equation  of  motion  becomes  by  (245i8) 

509.  When  there  is  no  external  force,  this  equation  becomes 

Dts'  ==:  —  as' v' ', 


the  integral  of  which  is 

102:/==^  —  av . 


'o 


in  which  A  is  an  arbitrary  constant.     Another  integration  gives 
C  av  —  A         C(j^       av  —  A\         C(      av—A\ 

in  which  c  is  the  Naperian  base,  and  q  the  radius  of  curvature  of 
the  fixed  line. 

35 


—  274  — 

510.  If  the  fixed  Hue  is  the  involute  of  the  circle,  and  if  its 
equation  is 

the  equation  (273o8)  becomes 

in  which  B  is  an  arbitrary  constant. 

511.  If  the  fixed  line  is  the  logarithmic  spiral,  and  if  its  equa- 
tion is 

i)=:  Re       , 

the  equation  (27328)  becomes 

a-\-b  '         ' 

in  which  B  is  an  arbitrary  constant. 

512.  If  the  fixed  line  is  the  cycloid,  and  if  its  equation  is 

Q  =  4  7?  sin  I' , 
the  equation  (27328)  becomes 

I  =  .,.     [a sni V  —  cos v)  c  +  -" 

in  which  B  is  an  arbitrary  constant. 

513.  When  the  resistance  of  the  line  is  constant,  and  the  resisting 
medium  is  moving  ivith  an  uniform  velocity  in  an  invariable  direction,  and 
the  resistance  arising  from  it  is  proportional  to  the  velocity  in  the  medium,  let 

a  be  the  constant  resistance  of  the  hne, 

h  the  resistance  of  the  medium  for  the  unit  of  velocity,  and 

I)  the  velocity  of  the  medium, 


—  275  — 

and  if  the  direction  of  the  motion  of  the  medium  is  assumed  for 
that  of  the  axis  of  x,  the  equation  of  motion  becomes 

Z>,/  =  Z>j,  12  —  a  —  lis'x  cos  \ 

in  which  it  is  carefully  to  be  observed  that  the  sign  of  a  must  be 
reversed  simultaneously  with  the  direction  of  motion. 

514.      When  the  fixed  line  is  straight  and  there  is  no  external  force 
the  integral  of  the  equation  (2755)  becomes 

log(/  —  5  cos ^  -|-^j  =:  ^  —  ht 

in  which  A  is  an  arbitrary  constant.     When 

a<^hh  cos  \ , 

the  velocity  of  the  body  will  never  be  destroyed,  but  will  constantly 
approximate  to 

0  cos^  — J. 

But  when 

a'^  bh  cosl, 

the  velocity  will  vanish  after  the  time  t^,  determined  by  the  equation 

log  (^  — b  cos  i):=A  —  hto. 

If  the   initial   velocity  of  the    body  had  been  negative,   the 
equation  of  motion  would  have  assumed  the  form 

log(— /+^cos^  +  ^^)  =  — yl  +  /^/; 

so  that  the  velocity  would  have  vanished  after  the  time  4,  deter- 


—  276  — 
mined  by  the  equation 

The  body  would  then  have  remained  at  rest  unless  the  con- 
dition (275i4)  had  been  satisfied,  in  which  case  its  subsequent  motion 
would  be  defined  by  the  equation  (275ii). 

515.  When  a  heavy  hody  moves  upon  a  fixed  straight  line,  and  the 
resistances  consist  of  a  ^constant  resistance,  arising  from  the  friction  along 
the  line,  and  also  of  a  resistance  arising  from  a  resisting  medium,  which 
has  a  uniform  motion  in  the  direction  of  the  fixed  line  ;  and  ivhen  the  re- 
sistance of  the  medium  is  proportional  to  the  square  of  the  velocity  of  the 
body  in  the  medium,  let 

a  be  the  constant  of  friction, 

h  the  velocity  of  the  medium,  and 

h  the  resistance  of  the  medium  for  the  unit  of  velocity. 

The  line  may  be  assumed  to  be  vertical  without  diminishing 
the  generality  of  the  investigation  and  the  equation  of  motion 
will  be 

D,s'  =g  —  a  —  h{s'  —  h)\ 

in  which  the  signs  of  a  and  h  must  be  reversed  simultaneously  with 
those  of  /  and  [s'  —  h)  resj)ectively.  The  equation  of  motion  has 
precisely  the  same  form  with  that  of  §  507,  so  that  the  forms  of 
the  integral  are  the  same  which  are  there  given,  but  the  constants 
are  not  subject  to  the  restrictions  of  that  section. 

If,  then,  the  initial  velocity  is  upward  and  exceeds  that  of 
the  medium,  when  the  medium  is  also  moving  upwards,  the  ascend- 
ing velocity  decreases  by  the  law  expressed  in  the  equation 

/_}  =  ^?^tan[(<-<r)V(M<7  +  «))], 


—  277  — 

in  which  t  is  an  arbitrary  constant.  This  law  of  ascent  continues 
until  the  body  is  brought  to  rest  when  the  medium  is  not  moving 
upwards.  But  when  the  medium  is  moving  upwards,  it  continues 
until  the  instant  (t),  when  the  velocity  of  the  body  is  the  same 
with  that  of  the  medium.  After  this  instant,  the  velocity  de- 
creases by  the  law 

^'-*  =  \/^T"°[(<-^)V(/'(i/  +  «))]; 

which  continues  forever  if 

and  the  velocity  constantly  approximates  to  that,  which  is  deter- 
mined by  the  equation 

But  when 

g-\-a>hP, 

the  body  is  brought  to  a  state  of  rest,  in  which  it  continues  per- 
manently if 

g  —  a  <ihP. 

But  if  the  motion  of  the  medium  is  upward,  and 

g  —  a>hb'^, 

the  body  moves  from  the  state  of  rest  with  an  increasing  descending 
velocity  of  which  the  law  is  expressed  by  the  equation 

in  which  t^  must  be  determined  so  that  the  instant  of  rest  coincides 


—  278  — 

with  that  given  by  the  equation  (2778).  The  increasing  velocity 
continually  approximates  to  that  which  is  determined  by  the 
equation 

g^a  =  h{s  —hf. 

Tlie  state   of  rest  to  which  the  body  is   brought,  when   the 
medium  is  not  moving  upwards,  is  permanent  if 

a  — g'^hh'^. 

But  if,  on  the  contrary, 

a  —  g  "^hb"^ 

the  body  moves  from  the  state  of  rest  with  an  increasing  descending 
velocity,  of  which  the  law  is  expressed  by  the  equation 

s' _  }  =  y/ ?=i^  tan  [(i;  -  r,)  y/ (A  (^  -  «) )]  , 
when 

in  which  Ti  must  be  determined  so  that  the  instant  of  rest  coincides 
with  that  given  by  the  equation  (27630).  This  law  of  motion  con- 
tinues until  the  instant  Tj,  when  the  downward  velocity  of  the 
body  becomes  the  same  with  that  of  the  medium;  and  after  this 
instant,  the  law  of  increasing  velocity  of  descent  is  expressed  by 
the  equation  (27729) ;  so  that  the  velocity  continually  approximates 
to  that  which  is  determined  by  the  equation  (278^). 

But  when  the  body  begins  to  descend  from  the  state  of  rest, 
and 


—  279  — 
the  law  of  descent  is  expressed  by  the  equation 

SO  that  the  increasing  velocity  constantly  approximates  to  that 
which  is  determined  by  the  equation 

a—g  =  h(s—hf. 

If  the  initial  velocity  is  downward,  and   exceeds  that   deter- 
mined by  the  equation  (2784),  the  decreasing  velocity  when 

is  expressed  by  the  equation 

s'-b  =  ^i^Goi\_{t-r)sj(h(g-a))-], 

in  which  t  is  an  arbitrary  constant.  If,  therefore,  the  motion  of 
the  medium  is  downward,  or  if  it  is  upward  and  the  condition 
(27724)  is  satisfied,  the  decreasing  velocity  continually  approximates 
to  that  which  is  determined  by  the  equation  (27729).  But  if  the 
motion  of  the  medium  is  upward  and  the  condition  (2772i)  is 
satisfied,  the  body  is  brought  to  a  state  of  rest  which  is  permanent 
if  the  condition  (277];)  is  also  satisfied.  If,  however,  the  condition 
(277ii)  is  satisfied  by  the  upward  motion  of  the  medium,  the  body 
leaves  the  state  of  rest  and  ascends  with  an  increasing  velocity, 
which  is  defined  by  the  equation 

s'-i  =  y'-^'Cot[(<-r,)v/(A(y  +  «))], 

in  which  Tj  must  be  determined  so  that  the  instant  of  rest  coin- 
cides  with   that   which   is    given   by  the  equation  (279i5).      The 


—  280  — 

ascending  velocity  continually  approximates  to  that  which  is 
determined  by  the  equation  (277i5). 

If  the  initial  velocity  is  downward,  and  exceeds  that  of  the 
medium,  when  the  medium  is  also  moving  downwards,  the  de- 
scending velocity,  when 

decreases  by  the  law,  expressed  in  the  equation 

in  which  t  is  an  arbitrary  constant.  This  law  of  descent  continues 
until  the  body  is  brought  to  rest,  when  the  medium  is  not  moving 
downwards;  but  when  the  medium  is  moving  downwards,  the 
law  continues  until  the  instant  t,  when  the  velocity  of  the  body 
is  the  same  with  that  of  the  medium.  After  this  instant,  the  law 
of  decreasing  velocity  becomes 

^'-b  =  ^''-^T<^n  [(t: - 0  V  (/* (« -9))] , 

which  continues  until  the  body  is  brought  to  rest,  when  the  condi- 
tion (2789)  is  satisfied.  But  when,  on  the  contrary,  the  condition 
(27812)  is  satisfied,  the  body  continues  to  move  forever  with  the 
law  of  decreasing  velocity  expressed  in  (28O19),  and  the  velocity 
continually  approximates  to  that,  which  is  determined  by  the 
equation  (2797).  When  the  body  has  been  brought  to  the  state 
of  rest,  the  condition  and  laws  of  leaving  it  are  the  same  with 
those  defined  in  (27923_3j),  when 


—  281 


THE     SIMPLE     PENDULUM     IN     A     RESISTING     MEDIUM. 

516.  When  the  curve  is  the  circwnfe?'enee  of  a  vertical  circle,  the 
problem  is  that  of  the  simple  pendulum  in  a  resisting  medium.  If  the  arc 
of  vibration  is  supposed  to  be  so  small  that  its  powers,  tvhich  are  higher 
than  the  square  mag  be  neglected,  and  if  the  7'csistance  arising  from  the 
medium  is  supposed  to  be  proportional  to  the  velocity,  and  to  he  combined 
zvith  a  constant  friction,  let 

a  be  the  friction,  and 

h  the  resistance  of  the  medium  for  the  unit  of  velocity, 

and   the  equation   of  motion  becomes,  by   adopting   the  notation 
of  §  487, 

in  which  the  sign  which  precedes  a,  must  be  the  reverse  of  that  of 
Dt  (p .     The  integral  of  this  equation  is 


in  which 


I     -^«     I     (To      — h^^     '       7J 


k  =  ^  j^cosa 


^  h  =  \/ ^sina. 


and  the  arbitrary  constants  have  been  determined  so  that  the  initial 
angular  velocity  (9J,)  shall  be  the  maximum  velocity,  and,  therefore, 
the  initial  value  of  y  is 

I    S  a 
—    9 
36 


—  282  — 

517.     The   equation  (28I21)  only  applies  to  the  first  vibration 
and  for  the  {m-\-  1)''  vibration,  the  correct  equation  is 


9 


=  ±^  +  fc-^^^'-'-hmkit-T„), 


in  which  t,„  is  the  instant  of  the  maximum  angular  velocity  (ff^  of 
that  vibration  and  the  doubtful  sign  is  alternately  positive  and 
negative  for  the  successive  oscillations,  so  that  the  position  of 
maximum  velocity  is  always  upon  the  descending  portion  of  the 
oscillation. 

518.  The  angular  velocity  of  vibration  is  expressed  by  the 
equation 

/ /      —^h{t  —  T„.)  cos  [k  (t  —  T„.)  -f  «] 

r  — 9m  c  COS  a  ' 

and  it  vanishes  for  the  instants 

which  correspond  to  the  beginning  and  end  of  the  oscillation.  The 
whole  time  of  oscillation  is,  therefore, 

T=-T^7i  i    -  sec  ct, 
k  \  (/ 

which  is  invariable,  although  it  exceeds  the  time  of  vibration  in  a  vacuum, 
in  consequence  of  the  factor,  sec  a . 

519.  The  angular  deviations  of  the  pendulum  from  the  verti- 
cal at  the  beginning  and  end  of  the  oscillation  are  given  by  the 
equation 

—  Ra  —    ,      I  R     (a-\-\n)  tan  a 


—  283  — 

whence  the  whole  arc  of  the  {m  -\-  1)''  vibration  is 

■w  ^     f      IR     « tan  a  ^       /i       ,  \ 

v^^l^>^d  -c  Cos  (1 71  tan  «). 

520.  The  angular  deviations  of  the  pendulum  from  the  ver- 
tical at  the  end  of  one  vibration  and  the  beginning  of  the  next  are 
identical,  but  the  deviation  from  the  point  of  maximum  velocity  is, 
on  account  of  the  change  in  the  position  of  this  point,  diminished 
by  the  quantity 

2Ra 
9 

The  successive  values  of  the  maximum  velocity  are  therefore 
connected  by  the  equation 

or 

f  ,     —  71  tan  a        ^        /  ^    —  (a-\-\Ti)  tan  a 

(pm  +  i  =  fpmC  _2«y/-c     V   -r.   y 

The  general  expression  for  the  maximum  velocity  is  then 
found  to  be 

/■  ,    — 7n7Ztana        o        /^    — («4- i;r)  tan  «  /c  —  '"7r  tana  — 1\ 

which,  on  account  of  the  smallness  of  a  and  a,  may  be  reduced  to 

/  /■     —  m  7Z  tan  «        r>  /  R 

fpm  =  ^>oO  — 2maU -. 

The  corresponding  value  of  the  arc  of  vibration  is 

<P,n^*2^o^  c  CosfiTTtana) ; -. 

or 

TT  tan  a  ima  R 


*P,„ 9^0  ^ 


—  284  — 

The  laio  of  the  diminidion  of  the  arc  of  vibration  and  of  the  maxi- 
mum of  velocity  is,  therefore,  such  that  either  of  these  quantities  consists 
of  two  terms,  one  of  ivhich  is  dependent  upon  the  portion  of  the  resistance, 
which  is  proportional  to  the  velocity,  and  decreases  in  geometrical  ratio, 
while  the  other  is  principally  dependent  upon  the  constant  friction  and  de- 
creases, sensibly,  in  arithmetical  ratio.  The  vibration  ceases  when  the  second 
term  of  either  of  these  quantities  surpasses  the  first. 

521.  If  the  resistance  is  proportional  to  the  square  of  the  velocity, 
and  if  h  is  its  value  for  the  unit  of  velocity,  the  equation  of  the 
motion  of  the  pendulum  is 

J}f(p  =  —  j^sm(p—h{I),cpf. 

If  one  of  the  first  integrals  of  this  equation  is  supposed  to  be 
(25426),  in  which,  however,  II  is  not  constant  but  variable,  the 
differential  of  (25426)  gives,  by  means  of  this  equation  and  (25426), 

n,ir=E''D,(pD'fcp-{-yBsm(pD,(p  =  —  hIl^{I),(pf 

=  —  2hl)t(p  (yB  cos  (p  -\-  IT) , 
I)^II=—2yhEcos(p  —  2hIl; 

and  the  integral  of  this  last  equation  if 

isbnfi=i2h, 
is 

II:=:zAe~^      '*  —  g R &\r\ ^i ^\n  {(f -\- ii) , 

in  which  A  is  an  arbitrary  constant.     The  equation  (25426)  is  then 
reduced  to 

R''  {D.iff  =z2Ac~'^^^''^  +2g  Rcos^i  cosilp  +  ^) ', 


of  which  the  integral  is 


J(t> 


E 


^  \/  [2  Ac  — <l>^^ /J' -{-2 g  Ji cos lA. cos  ((p-\- ia)^' 


—  285  — 

The  signs  which  precede  the  quantities  h  and  ^i  must  be  re- 
versed in  the  alternate  oscillations. 

522.  The  angle  of  greatest  deviation  from  the  vertical  for 
the  (m  -f-  l)st  oscillation  is  determined  by  the  equation 

—  (fm  tan  u         .  V 

c     ^        ^cos(9)^  — /^,) 


g  a  COS  jt* 

If  J  is  adopted  as  the  sign  of  finite  differences,  this  equation 
gives,  when  fi  is  so  small  that  its  square  may  be  neglected, 

J  [cos  (p„,  —  (sin  9,„  —  (p^  cos  (p^)  fi']  =  2  (sin  (p^  —  (p^  cos  (p„,)  ^ . 

When  the  oscillations  of  the  pendulum  are  so  small  that  the 
fourth  power  of  y,„  may  be  neglected,  and  also  the  product  of  jit 
by  fpm  ^  ^my  this  cquatiou  is  reduced  to 

of  which  the  approximate  integral  is 

qPo 

523.     The  substitution  of  (2855)  reduces  (28427)  to  the  form 


E 


{D.cpf  =  cos  {cp  +  .a)  - c-  ^'^  +  ^'^"^ '"" ^ cos  (9,, - ^) , 


2  g  cos  [I 
which,  when  in  is  so  small  that  its  square  may  be  neglected,  becomes 

7? 

—  (A  9)2=  cos  {(p+fi)  —  COS  (9),,  —  ^)  +  COS  (p„,  {(p  +  (p,n)  u. 

=  COS  9)  —  COS  (p„,  —  in  [sin  9  +  ^^^  ^m  —  (^  +  ^m)  cos  f/)„,] . 
When  the  oscillations  are  very  small,  this  equation  may  be 


—  286  — 
still  further  reduced  to 

which  gives 

The  integral  of  this  equation  is 

<f  =  ,y„sm[y'|(^-r)-J-^^V/«_y^)]. 

The  time  of  the  descending  semioscillation,  deduced  from  this 
equation,  is 

The  time  of  the  preceding  semioscillation  is  obtained  by  re- 
versing the  sign  of  fn,  which  gives 

and  the  time  of  the  ivhole  oscillation  is,  therefore,  the  same  as  if  the  pen- 
dulum vibrated  in  a  vacuum.  The  preceding  formulae  and  conclusions 
coincide,  substantially,  with  those  which  are  given  by  Poisson. 

524.     If  the   law   of  the   resistance  to   the   motion  of  the  pendulum 
may  he  expressed  as  a  function  of  the  time,  let 

ST  denote  the  resistance, 

and  the  motion  of  the  pendulum  in  a  small  arc  is  expressed  by 
the  formulce  (26O9)  and  (261;).     If  ^  is  a  periodic  function,  which 


—  287  —  . 

has  the  same  period  with  that  of  the  vibration  of  the  penduhim, 
it  may  be  expressed  in  the  form 

gr= /,,  +  2  1, [/.,  cos  (z  yi  + /^,)]  ; 

and,  if  the  variable  portions  of  A  sin  b  and  A  cos  b  are  denoted  by 
d,  these  equations  give 

^d(^sin})=/i„(l-cos(i;y/|))— /,,<y/|sin/:i,— iA.cos(2('y/|+/:),) 

+  ,^  sin  ((,•  +  !)  y  I  + /J,)-^^  sin  ^,-] ; 

which  vanish  with  t. 

525.  i/^  the  mbrations  of  the  pendulum  cause  the  medium  to  oscil- 
late, the  ])eriod  of  the  oscillations  of  the  medium  is  frobably  the  same  ivith 
that  of  the  pendtdum,  but  the  successive  2)hases  of  the  motion  of  the  medium 
are  likely  to  lag  someivhat  behind  those  of  the  j^cndulum.  Hence  the 
reLitive  velocity  of  the  pendulum  to  the  medium  may  be  ex- 
pressed by  the  equation 

V=vAco^{tyJ^  +  b  +  i^). 

in  which  A  and  b  may  be  regarded  as  constant  for  a  single  vibra- 
tion. 


—  288  — 

If,   then,  the   resistance    of  the   medium   is  proportional  to  the 
relative  velocity,  the  value  of  ^  assumes  the  form 

3r=2/i^cos(/^|  +  5 +  /:?); 
and.  the  equations  (2878_i8)  give 

^^(^sin^)  =  -y|sin(5  +  /?) 

-^cos(2y|+i  +  /^)  +  ^cos(i  +  /i), 
£jd(^cos5)  =  -y|cos(J  +  « 

—  isin(2^fy/|+J+/^)  +  ^sin(Z.  +  /?), 
whence 

|^log^^_y|cos/i-|sin(2^y/|+25  +  /?)  +  ^sin(2^i  +  ^i), 
|,yj  =  _y|sin/?  — ^cos(2y|+2^  +  /:?)  +  ^cos(25  +  /^). 

If  T  is  the  time  of  vibration  of  the   pendulum,  the    changes 
of  A  and  h  in  a  single  vibration  are  given  by  the  formula 

z/ logil  =  — ^T^y/lcos/i  =  —  TT -cos  f^, 

^j  =  _^7^./|sin/S==  — 7i-sin/?. 

If  the  resistance   is  p?vpo)iional  to  the  square   of  the  velocity,  the 
value  of  ?r  assumes  the  form 

Sr=x  2/cA^  4-  2k A^ cos(2tJj^+2b  +  2/?), 

in  which  the  sign  of  k  must  be  reversed,  when  the  direction  of  the 


—  289  — 

relative   nirytion   of   the   body   to   the    medium   is  reversed.     This 
value  of  3°  gives 

^d(Jsm5)  =  2-2cos(<'y/|)  +  cos(y|+2i  +  2/i) 

—  j  cos  (3<i/|  +  2*  +  2/i)  —  I  cos  (24  +  2  (i), 

^,d  {A  cosi)  =  -  2  sin  (<y/|)  -  sin  (^ |  +  25  +  2  /?) 

-Jsin(3;y/|  +  25  +  2/i)  +  |sin(2i4-2/?); 
whence 

^^,(5^^  =  2sinZ.-2sin(/y/|  +  ^)4-sin(y|+^  +  2/i) 

—  |sin(3/y/|  +  3^4-2/i)  +  isin(3^  +  2/^)— sin(^  +  2/^), 

^^(y^=:2cosi-2cos(y|  +  ^)  +  cos(y|  +  J  +  2r^) 

—  |cos(3/y/|  +  3^i  +  2fi)  +  icos(3^  +  2/i)  — cos(/i4-2^). 

The  changes  of  ^  and  h  in  a  vibration  are  found,  by  having 
regard  to  the  reversal  of  the  sign  of  k  which  corresponds  to  that 
of  V,  to  be 

g  J  h  =^  —  -J/  Ji;  j[  sin  (j . 

Jf  the  law  of  the  resistance  is  similar  to  that  of  friction  so  as  to 
be  constant  if  the  medium  is  at  rest,  it  must,  when  the  medium  is  in 
motion,  be  proportional  to  the  quotient  of  the  relative  motion  of 
the    body   through   the    medium    divided    by  the    velocity    of  the 

37 


—  290  — 
body.     The  form  of  ST  is,  then, 

gr_«  cos  (/y/l +  &  +  /?) 
—       cos{ts/i  +  b) 

in  which  the  sign  of  a  must  be  reversed,  when  the  direction  of  the 
relative  motion  of  the  body  to  the  medium  is  reversed.  This  value 
of  ?r  gives 

-^()'(^sinZ>)i3::  — cos(^'i/|4-fj)  — sin/Jcosnogtan(i:T+^^^^+^), 

^d{A  cos  b)=^  —  sin  (/y/|  +  /i)  +  sin/^sini  log  tan(i  n  +  ^-^^|i-^)  ; 
whence 

d  {  =  -  ^  cos  (y  I  +  i  +  (i)  - 1^  sin /i  log  tan  (i  ^  +  (vi+_*) . 
The  changes  of  A  and  b  in  a  vibration  are 

J  b:=  -T-sm(i  log  tan  i  /i? . 

The  combination  of  these  values  give 
^^^_^_;r^'-cos/^  — -V---^'cosf^, 
J  b=z~sm(-i  lopj  tan  i  /:?  —  n  -  sin  /•>  —  -U  -  ^  sin  f:/ . 

The  change  of  ^  is  exhibited  in  the  motion  of  the  pendulum 


—  291  — 

by  a  change  in  the  time  of  vibration,  which  differs  from  that  Avhich 
it  would  be  in  a  vacuum.     The  diflerence  is 

y    9  n 

526.  The  vibration  of  the  pendulum  may  be  regarded  as 
affected  by  the  medium  not  only  in  consequence  of  its  direct  action 
as  resistance,  but  also  indirectly,  because  a  portion  of  the  medium 
may  be  regarded  as  composing  a  part  of  the  moving  body,  and  its 
motion  is  sustained  by  the  action  of  gravitation  upon  the  body. 
If,  then, 

q  denotes  the  ratio  of  the  mass  of  that  portion  of  the  medium 
which  moves  with  the  body  to  the  mass  of  the  body, 

the  motion  of  q  may  be  assumed  to  have  a  period  identical  with 
that  of  the  body,  and  an  amplitude  of  excursion  proportional  to 
that  of  the  body,  so  that  its  velocity  may  be  of  the  form 

r  =  A' J  y/l  cos  (y  !  +  «-/}'). 

The  resistance,  then,  arising  from  the  preservation  of  the 
motion  of  q  may  be  expressed  in  9°  by  the  form 

The  similarity  of  this  form  to  that  of  (2884)  shows  that  the 
corresponding  influence  upon  A  and  h  may  be  expressed  by  the 
equations 

z/log^r=z  — l^sin/f, 


—  292  — 

The  importance  of  this  form  of  resistance  was  first  noticed 
by  DuBUAT  and  has  been  investigated  experimentally  by  Dubuat, 
Bessel,  and  Baily.  The  formulae  (29O27)  and  (29I29)  may  be  adopted 
as  a  guide  in  the  conduct  of  these  and  similar  investigations. 

527.  In  the  application  of  the  preceding  formuloB  to  the  re- 
duction of  experiments,  the  quantities  a^  h,  Jc,  and  q  are  inversely 
proportional  to  the  density  of  the  body,  and  directly  proportional 
to  the  density  of  the  medium,  and  for  bodies  of  similar  forms  they 
are  nearly  in  an  inverse  ratio  to  their  linear  dimensions.  For 
pendulums  of  different  lengths,  k  is  proportional  to  the  length  of 
the  pendulum,  and  h  to  the  time  of  vibration.  If  H^  denotes  the 
resistance  of  the  medium  which  is  proportional  to  the  velocity  for 
the  unit  of  weight  and  the  unit  of  surface,  and  if  ^  denotes  the 
resistance  which  is  proportional  to  the  square  of  the  velocity  for 
the  same  unit  of  weight  and  surface,  the  values  of  h  and  k,  for  the 
units  of  weight  and  surface,  are 

/;  —  —  TTT 
k  =  \gRII^, 

528.  The  best  experiments  which  have  been  made  with  the 
pendulum  are  almost  wholly  free  from  any  constant  term  of  resist- 
ance, so  that,  in  their  discussion,  this  term  may  be  neglected  which 
reduces  the  formula  (290.26)  to  the  form 

AA  =  —  \  TH^Aco^(i  —  ^  RH^A^co^(i, 

of  which  the  approximate  integral  is 

1°=  (1  +  8-TO:)  -  1«S  (1  +  s-^)  =  J '»  TII,  cos  t.  ■ 

529.  In  order  to  illustrate  these  formulas,  they  may  be  ap- 
plied to  some  of  the  experiments  which  have  been  actually  made, 


—  293  — 

and  in  which  the  diminution  of  the  arc  of  vibration  has  been  ob- 
served. For  this  purpose  the  observations  of  Newton,  Dubuat, 
BoRDA,  Bessel,  and  Baily  are  selected,  and  the  formula  (29228)  is 
found  to  be  applicable  to  all  these  experiments,  although  the  values 
of  Hi  and  H2  are  different  for  the  different  experiments.  The  unit 
of  length  which  is  here  adopted  is  the  meter,  the  unit  of  weight 
is  the  chiliogramme,  and  that  of  time  is  the  mean  solar  second. 
The  measures  and  weights  are,  however,  given  in  the  form  in 
which  they  were  actually  observed. 

530.  In  Newton's  first  series  of  experiments  upon  the  dimi- 
nution of  the  oscillations  of  a  pendulum,  a  wooden  sphere  of 
61  English  inches  in  diameter,  weighing  57 2^?  ounces,  of  about 
0.56  specific  gravity,  and  susjDcnded  by  a  fine  wire  so  as  to  give 
10 i  feet  for  the  length  of  the  pendulum,  was  vibrated  until  the  arc 
of  descent  was  diminished  one  fourth  or  one  eighth  of  its  initial 
extent,  and  the  number  of  vibrations  was  recorded.  From  the  re- 
duction of  these  observations,  I  have  obtained  for  the  values  of 
11^  and  II2 

^1=0.0223  sec/?, 

7/2  =0.4473  sec/?. 

In  Newton's  second  series  of  experiments,  a  leaden  sphere  of 
2  inches  in  diameter,  weighing  261  pounds,  and  suspended  so  as  to 
give  10?  feet  for  the  length  of  the  pendulum,  was  vibrated  in  the 
same  way  as  in  the  former  series.  From  the  reduction  of  these 
observations,  I  have  obtained 

^1=  0.2044  sec /?, 
^2  =  0.701  sec/?. 

To  test  the  accuracy  of  these  reductions,  and  their  conformity 
with  the  given  observations,  I  have  computed  the   lengths  of  the 


294 


observed  arcs  of  vibration,  and  have  placed  them  in  the  following 
table  for  comparison. 


COMPARISON     OF    NEWTON  S     EXPERIMENTS    UPON    VIBRATIONS     OF    THE     PENDULUM 

WITH    COMPUTATION. 


WOODEN 

SPHERE. 

LEADEN 

SPUERE. 

m 

Computed 

Observed 

C      0 

m 

Computed 
An 

Observed 

C     0 

in. 

in. 

in. 

in. 

in. 

in. 

0 

64.08 

64 

.08 

0 

64.03 

64 

.03 

H 

56.02 

bQ> 

.02 

30 

56.04 

56 

.04 

22f 

47.91 

48 

—.09 

70 

47.92 

48 

—.08 

0 

31.86 

32 

—.14 

0 

31.92 

32 

—.08 

181 

27.92 

28 

—.08 

53 

28.00 

28 

0. 

41f 

24.19 

24 

.19 

121 

24.07 

24 

.07 

0 

15.99 

16 

—.01 

0 

16.01 

16 

.01 

35i 

14.01 

14 

.01 

901 

13.99 

14 

—.01 

83i 

11.99 

12 

—.01 

204 

11.99 

12 

—.01 

0 

8.04 

8 

.04 

0 

8.05 

8 

.05 

69 

7.01 

7 

.01 

140 

7.01 

7 

.01 

lG2i 

5.95 

6 

—.05 

318 

5.95 

6 

—.05 

0 

4.01 

4 

.01 

0 

4.03 

4 

.03 

121 

3.50 

U 

0. 

193 

3.49 

H 

—.01 

272 

2.99 

3" 

—.01 

420 

2.97 

3 

—.03 

0 

1.98 

2 

—.02 

0 

2.04 

2 

.04 

1G4 

1.74 

H 

—.01 

228 

1.74 

If 

—.01 

374 

1.52 

H 

.02 

518 

1.46 

l| 

—.04 

0 

1.00 

1 

0. 

226 

.88 

1 

0. 

510 

.75 

1 

0. 

With  these  values  of  Hi  and  II2,  a  minute  arc  of  vibration 
of  the  wooden  sphere  would  be  reduced  one  eighth  part  in  446 
vibrations,  and  one  fourth  part  in  961  vibrations,  and  a  minute 
arc  of  vibration  of  the  leaden  sphere  would  be  reduced  one  eighth 
part  in  290  vibrations,  and  one  fourth  part  in  625  vibrations. 

531.  DuBUAT  vibrated  in  water  a  sphere  of  2.645  French 
inches  in  diameter,  weighing  in  air  40068  grains,  and  in  water 
36448  grains,  and  suspended  so  that  the  length  of  the  pendulum 


—  295  — 

was  36.714  inches;  he  observed  the  arc  of  descent  at  each  succes- 
sive oscillation.  From  these  observations,  I  have  obtained  a  result 
which  corresponds  with  his  own  in  respect  to  the  law  of  diminution 
of  oscillation,  and  which  gives  for  the  values  of  II^  and  11^  in  water 

^2  =378.7  sec/?' 

DuBUAT  also  vibrated  in  air  a  paper  sphere  of  4.0416  inches  in 
diameter,  weighing  in  air  155  grains,  with  a  density  11.33  times  as 
great  as  that  of  air,  and  suspended  by  a  fine  thread  so  that  the 
length  of  the  pendulum  was  36.714  inches.  From  these  observa- 
tions, I  have  deduced 

^2  =  0.37  sec /i. 

The  following  table  contains  the  comparison  of  Dubuat's  experi- 
ments with  the  computations  derived  from  the  values  of //i  and  H^, 

COMPARISON    OF    DUBrAX's    EXPERIMEXTS    UPON    THE    DIMINUTION   OF    THE    ARC    OF 
VIBRATION    OF    A    PENDULUM    WITH    COMPUTATION. 


SPHERE  IN  WATER. 

SPHERE  IN  AIR. 

m 

Computed 

Observed 
An 

1 

C     0 

m 

Computed 

A. 

Observed 

C       0 

in. 

in. 

in- 

in. 

in. 

in. 

0 

12.00 

12.00 

0. 

0 

11.90 

12.00 

—.10 

1 

9.21 

9.25 

—.04 

1 

10.10 

lO.dO 

.10 

2 

7.47 

7.42 

.05 

2 

8.77 

8.70 

.07 

3 

6.28 

6.25 

.03 

3 

7.75 

7.79 

—.04 

4 

5.42 

5.33 

.09 

4 

6.94 

6.96 

—.02 

5 

4.77 

4.75 

.02 

6 

4.25 

4.25 

0. 

7 

3.84 

3.83 

.01 

8 

3.50 

3.48 

.02 

9 

3.22 

3.23 

—.01 

10 

2.97 

2.98 

—.01 

_  296  — 

532.  BoRDA  vibrated  a  platinum  sphere  of  16 J  lines  in  diameter, 
weighing  with  the  wire  and  screw  9963  grains,  and  suspended  by  a 
wire  so  that  the  length  of  the  pendulum  was  3.95497  metres.  These 
observations  give  for  the  values  of  Hi  and  11^  in  air 

^1=:  0.10722  sec/:?, 

^,r=  0.6267  sec/:?. 

In  his  observations  for  determining  the  length  of  the  seconds 
pendulum,  this  same  pendulum  was  vibrated  by  Borda,  and  the 
lengths  of  its  arcs  of  vibration  were  observed.  From  the  mean  of 
these  observations,  I  have  obtained  the  values  of  H^  and  II^, 

//i=  0.11214  sec /i, 
^2  =0.6564  sec /i. 

Borda  vibrated  the  same  sphere  with  a  smaller  wire,  so  that 
the  weight  was  reduced  to  9958  grains,  and  the  length  increased  to 
3.95597  metres.     From  these  observations  I  have  derived 

^1  =  0.1134  sec /^, 
7/2  =  0.590  sec /^ 

The  comparison  of  Borda's  experiments  with  the  computations 
based  upon  these  values  of  H^  and  H^  is  contained  in  the  following 
tables. 


COMPARISON    OF    BORDA's     OBSERVATIONS    UPON     THE     DIMINISHED    VIBRATIONS    OF 
THE    PENDULUM    WITH    COMPUTATION. 

First  Experiment  icith  direct  re/cnnce  to  the  Dixiinution  of  the  Arc  of  Vibration. 


m 

Computed 

Observed 
An 

C       0 

m 

Computed 

Observed 
An 

C     0 

0 

120'0 

120'0 

0. 

12600 

4.2 

4.1 

1 

0.1 

1800 

61.2 

61.2 

0. 

14400 

2.8 

2.7 

3  GOO 

35.6 

35.4 

.2 

1 6200 

1.9 

1.8 

5400 

22.1 

21.9 

.2 

18(100 

1.3 

1.2 

7200 

14.2 

14.1 

.1 

19800 

0.9 

0.8 

9000 

9.4 

9.4 

0. 

2 1 600 

0.6 

0.5 

10800 

6.2 

6.3 

—.1 

36000 

0.002 

Very  minute. 

—  297  — 


Experiments  for  determining  the  Length  of  the  Second's  Pendtihtm  with  the  Pendulum  used  in  the  First 

Experiment. 


m 

Mean  Value. 

Computed 
An 

Observed 

^,n 

a—0 

Computed 

Observed 
An 

C—0 

Computed 

Observed 

C—0 

0 

d 

64 

1 

0 

67 

m 

0 

63' 

63' 

/ 
0 

2169 

321 

32 

1 

2 

34 

34 

0 

32 

32 

0 

4338 

18' 

19 

—  1 

181 

19 

\ 
2 

18 

18 

0 

6507 

lOi 

11 

1 

11 

11 

0 

101 

11 

2 

8676 

6l 

7 

-\ 

61 

7 

-i 

6 

6 

0 

0 

60 

60 

0 

61 

61 

0 

64^ 

641 

0 

2169 

31 

31 

0 

31^ 

311 

0 

321- 

321 

0 

4338 

171 

17 

1 

171 

18 

i 

2 

18 

171 

2 

6507 

10 

10 

0 

10 

10 

0 

10^ 

10 

i 

8676 

6 

6 

0 

6| 

6 

1 
2 

0 

63 

63 

0 

68 

68 

0 

61 

61 

0 

2169 

32 

32 

0 

34 

341 

1 
— 2 

31  ^ 

311 

0 

4338 

18 

18 

0 

19 

191 

1 
— 2 

171 

17 

\ 

6507 

lOi 

10 

i 

11 

lU 

_L 

10 

10 

0 

8676 

6 

6i 

J. 

4 

61 

7 

1. 

6 

6 

0 

0 

591 

591 

0 

571 

571 

0 

62 

62 

0 

2169 

301 

301 

0 

30' 

30 

0 

3U- 

311 

0 

4338 

17" 

17' 

0 

17 

17 

0 

18 

17 

1 

6507 

10 

10 

0 

91 

10 

-\ 

10^ 

10 

JL 
2 

8676 

6 

6 

0 

6 

6 

0 

0 

67 

67 

0 

65 

65 

0 

63 

63 

0 

2169 

34 

34 

0 

33 

331 

— i 

32 

32 

0 

4338 

181 

19 

1. 

181 

i8i 

0 

18 

171 

2 

6507 

11 

11 

0 

101 

11' 

—h 

101 

10 

J. 

2 

8676 

61 

6 

i 

61 

^\ 

0 

6 

6 

0 

0 

71 

71 

0 

591 

591 

0 

2169 

35 

341 

2 

31 

31 

0 

4338 

191 

19 

JL 

2 

171 

17 

h 

6507 

11 

11 

0 

10 

10 

0 

8676 

7 

7 

0 

6 

6 

0 

Experiments  for  determining  the  Length  of  the 

Second's  Pendulum  with  the  Second  Pendulum. 

m 

Comp'd 
An 

Observ'd 

C-0 

m 

Comp'd  ( 

Jbserv'd 
A,n 

G-0 

m 

Computed 
An 

Observed 
An 

G-0 

0 

551 

54 

0 

0 

79 

/ 

79 

0 

0 

111 

110 

1 
1 

1575 

341 

35 

I 

1538 

47 

47 

0 

1445 

641 

641 

0 

3150 

221 

23 

2 

3114 

30 

30 

0 

2970 

401 

40 

2 

4725 

15 

16 

—1 

4690 

191 

20 

0 

4495 

26 

26 

0 

6300 

10 

101 

1 
2 

6266 

13" 

14 

—1 

6020 

171 

18 

1 

2 

7875 

7 

H 

L 

2 

7842 

9 

H 

L 

7545 

12" 

12 

0 

9450  1     5 

5 

0 

9418 

5 

61 

i 

2 

9070 

8 

81 

1 

38 


—  298  — 

533.  In  Bessel's  experiments  made  for  the  determination  of 
the  length  of  the  second's  pendidum  of  Konigsberg,  a  brass  sphere 
of  24.164  hnes  in  diameter,  weighing  0^695364  was  suspended  so 
that  the  length  of  the  pendulum  was  1305.3  lines.  From  his  ob- 
servations with  this  pendulum,  I  have  found  these  values  of  Hi 
and  /^. 

//i  =  0.05698  sec /^, 
^^2=  0.529  sec /':J. 

The  same  sphere  was  also  vibrated  with  a  length  of  pendulum 
of  441.8  lines,  from  the  observations  of  which  I  have  deduced 

//i  =  0.0452  sec  f:f, 

^2  =0.587  sec/":?. 

Bessel  also  vibrated  an  ivory  sphere,  weighing  0M5112,  and 
having  a  diameter  of  24.094  lines,  with  each  of  the  preceding 
lengths  of  pendulum.  From  his  observations  with  this  sphere  and 
the  long  pendulum,  I  have  obtained 


//i  =  0.05517  sec  fi, 
7/2  =  0.512  sec  f:?; 


and  from  his  observations  w4th  the  short  pendulum, 

^1  ==0.0509  sec /^, 
7/2  =  0.282  sec  ^^. 

In  Bessel's  experiments  for  the  determination  of  the  length  of 
the  second's  pendulum  at  Berlin,  a  hollow  cylinder  was  vibrated, 
of  which  the  diameter  of  the  base  was  15.305  lines,  and  the  altitude 


_  299  — 

15.296  lines,  weighing,  with  its  appendages,  Avhen  it  was  filled  with 
lead,  0^67920,  and  when  it  was  empty,  0^22595.  It  w\ns  suspended 
in  two  different  modes,  in  one  of  which  the  length  of  the  pen- 
dulum w^as  1304.8  lines,  when  the  cylinder  was  filled,  and  1303.8 
lines,  when  it  was  empty  ;  and,  in  the  other  mode  of  suspension 
the  length  was  440.9  lines  when  the  cylinder  was  filled,  and 
440.7  lines  when  it  w^as  empty.  From  his  observations  with  this 
pendulum,  I  have  obtained  the  following  values  of  H^  and  i^. 

When  the  cylinder  was  full,  and  the  suspension  was  long,  the 
values  were 

^i:=  0.08544  sec /^, 
H^  =  0.733  sec  (i  ; 

when  it  was  full,  and  the  suspension  short,  they  were 

^i=zi  0.07026  sec  fi, 
^2  =.0.724  sec /^. 

When  the  cylinder  was  empty,  and  the  suspension  long,  the 
values  were 

^=0.09578  sec /'i, 
//=:  0.559  sec  (S  ; 

when  it  was  empty,  and  the  suspension  short,  they  w^ere 

^1=  0.07003  sec /i, 
^2  =0.270  sec/':?. 

In  order  to  compare  the  theory  of  these  values  w^itli  ex- 
periment, all  the  values  of  observation  have  been  recomputed, 
and  the  comparisons  are  contained   in  the  following  tables. 


—  300  — 


COMPARISON  OF  BESSEL's    OBSERVED  ARCS    OF  VIBRATION   OF   THE   PENDULUM  WITH 

THE  COMPUTED  ARCS. 


1.  Experiments  with  the  Brass  Sphere  and  Long  Suspension. 


m 

Computed 

Observed 

An 

C      0 

Computed 
-4m 

Observed 

C      0 

Computed 

Observed 
Am 

C     0 

0 

38.3 

38.3 

0 

39.0 

39.0 

0 

39.5 

39.5 

0 

500 

33.7 

33.8 

—.1 

34.2 

34.2 

0 

34.6 

34.6 

0 

1000 

29.7 

29.8 

—.1 

30.2 

30.1 

.1 

30.5 

30.5 

0 

1500 

26.4 

26.4 

0 

26.8 

26.8 

0 

27.1 

26.8 

.3 

2000 

23.5 

23.6 

—.1 

23.9 

23.8 

.1 

24.1 

23.9 

.2 

2500 

21.0 

20.9 

.1 

21.3 

21.3 

0 

21.6 

21.6 

0 

3000 

18.8 

18.8 

0 

19.1 

19.2 

—.1 

19.3 

19.3 

0 

3500 

16.9 

16.9 

0 

17.2 

17.2 

0 

17.4 

17.3 

—.1 

4000 

15.3 

15.4 

—.1 

15.5 

15.5 

0 

15.7 

15.7 

0 

0 

39.7 

39.9 

—.2 

39.0 

39.3 

—.3 

39.6 

39.7 

—.1 

500 

34.8 

34.6 

.2 

34.2 

34.1 

.1 

34.7 

34.8 

—.1 

1000 

30.7 

30.4 

.3 

30.2 

30.0 

.2 

30.6 

30.5 

.1 

1500 

27.2 

27.1 

.1 

26.8 

26.4 

.4 

27.1 

26.9 

.2 

2000 

24.2 

24.1 

.1 

23.9 

23.5 

.4 

24.2 

24.0 

.2 

2500 

21.6 

21.5 

.1 

21.3 

20.9 

.4 

21.6 

21.4 

.2 

3000 

19.4 

19.3 

.1 

19.1 

18.5 

.6 

19.4 

19.3 

.1 

3500 

17.4 

17.3 

.1 

17.2 

16.4 

.8 

17.4 

17.3 

.1 

4000 

15.7 

15.5 

.2 

15.5 

14.6 

.9 

15.7 

15.5 

.2 

0 

38.6 

38.6 

0 

40.0 

40.3 

—.3 

40.1 

39.9 

.2 

500 

33.9 

33.9 

0 

35.1 

34.9 

.2 

35.1 

35.2 

1 

1000 

29.9 

29.9 

0 

30.9 

30.8 

.1 

31.0 

31.0 

0 

1500 

26.5 

26.6 

—.1 

27.4 

27.2 

.2 

27.4 

27.5 

1 

2000 

23.7 

23.6 

.1 

24.4 

24.2 

.2 

24.4 

244 

0 

2500 

21.1 

21.2 

—.1 

21.8 

21.8 

0 

21.8 

21.9 

— 1 

3000 

19.0 

19.0 

0 

19.5 

19.5 

0 

19.6 

19.6 

0 

3500 

17.1 

17.1 

0 

17.5 

17.4 

.1 

17.6 

17.6 

0 

4000 

15.4 

15.4 

0 

15.8 

15.6 

.2 

15.8 

15.9 

— 1 

0 

39.1 

39.1 

0 

39.3 

39.2 

.1 

38.8 

38.5 

.3 

500 

34.3 

34.3 

0 

34.5 

34.5 

0 

34.0 

34.0 

0 

1000 

30.3 

30.3 

0 

30.4 

30.5 

—.1 

30.1 

30.2 

— 1 

1500 

26.9 

26.9 

0 

27.0 

27.1 

—.1 

26.7 

27.0 

—.3 

2000 

23.9 

23.9 

0 

24.0 

24.2 

.2 

23.8 

24.0 

—.2 

2500 

21.4 

21.4 

0 

21.5 

21.8 

—.3 

21.3 

21.5 

—.2 

3000 

19.2 

19.3 

—.1 

19.3 

19.4 

—.1 

19.1 

19.3 

—.2 

3500 

17.2 

17.3 

—.1 

17.3 

17.5 

—.2 

17.2 

17.4 

—.1 

4000 

15.5 

15.6 

—.1 

15.6 

15.7 

—.1 

15.5 

15.5 

0 

—  301 


1.  Expe\ 

•iments  wi 

th  the  Brass  Sphere 

and  Long 

Susj)ension.  —  Continued. 

m 

Computed 

A. 

Observed 

C     0 

Computed 
An 

Observed 

C—0 

Computed 

An 

Observed 
An 

C     0 

0 

39.1 

39.1 

0 

37.8 

37.7 

.1 

39.6 

39.7 

—.1 

500 

34.3 

34.2 

.1 

33.2 

33.3 

—.1 

34.7 

34.7 

0 

1000 

30.3 

30.2 

.1 

29.4 

29.3 

.1 

30.6 

30.6 

0 

1500 

26.9 

27.0 

—.1 

26.1 

26.2 

—.1 

27.1 

27.1 

0 

2000 

23.9 

24.0 

—.1 

23.2 

23.4 

—.2 

24.2 

24.1 

.1 

2500 

21.4 

21.5 

—.1 

20.8 

20.9 

—.1 

21.6 

21.6 

0 

3000 

19.2 

19.2 

0 

18.7 

18.7 

0 

19.4 

19.4 

0 

3500 

17.2 

17.3 

—.1 

16.8 

16.7 

.1 

17.4 

17.3 

.1 

4000 

15.5 

15.5 

0 

15.1 

15.2 

—.1 

15.6 

15.6 

.1 

0 

39.0 

39.0 

0 

41.7 

41.6 

.1 

39.4 

39.4 

0 

500 

34.2 

34.1 

.1 

36.5 

36.6 

—.1 

34.6 

34.5 

.1 

1000 

30.2 

30.1 

.1 

32.1 

32.3 

.2 

30.5 

30.4 

.1 

1500 

26.8 

26.5 

.3 

28.4 

28.6 

—.2 

27.0 

27.1 

— .1 

2000 

23.9 

24.0 

—.1 

25.3 

25.5 

.2 

24.1 

24.1 

0 

2500 

21.3 

21.4 

—.1 

22.5 

22.7 

.2 

21.5 

21.6 

—.1 

3000 

19.1 

19.2 

—.1 

20.2 

20.3 

—.1 

19.3 

19.4 

—.1 

3500 

17.2 

17.2 

0 

18.1 

18.1 

0 

17.3 

17.3 

0 

4000 

15.5 

15.5 

0 

16.3 

16.3 

0 

15.6 

15.6 

0 

0 

39.2 

39.4 

—.2 

38.6 

38.6 

0 

38.5 

39.3 

—.8 

500 

34.4 

34.4 

0 

33.9 

33.4 

.5 

33.8 

34.2 

—.4 

1000 

30.3 

30.3 

0 

29.9 

29.9 

0 

29.8 

30.0 

—.2 

1500 

26.9 

26.9 

0 

26.5 

26.7 

2 

26.5 

26.3 

.2 

2000 

24.0 

24.0 

0 

23.7 

23.6 

.1 

23.6 

23.2 

.4 

2500 

21.4 

21.5 

— 1 

21.1 

21.4 

—.3 

21.1 

20.5 

.6 

3000 

19.2 

19.2 

0 

19.0 

19.2 

2 

18.9 

18.3 

.6 

3500 

17.2 

17.1 

.1 

17.1 

17.2 

— 1 

17.0 

16.3 

.7 

4000 

15.6 

15.4 

.2 

15.4 

15.4 

0 

15.3 

14.5 

.8 

0 

40.0 

39.9 

.1 

39.9 

39.6 

.3 

39.3 

39.0 

.3 

500 

35.1 

34.9 

.2 

35.0 

35.0 

0 

34.5 

34.5 

0 

1000 

30.9 

30.9 

0 

30.8 

30.9 

—.1 

30.4 

30.5 

— 1 

1500 

27.4 

27.5 

— 1 

27.3 

27.5 

2 

27.0 

27.1 

— 1 

2000 

24.4 

24.4 

0 

24.3 

24.3 

0 

24.0 

24.1 

— 1 

2500 

21.8 

21.9 

— 1 

21.7 

21.7 

0 

21.5 

21.5 

0 

3000 

19.5 

19.7 

— 2 

19.5 

19.5 

0 

19.3 

19.3 

0 

3500 

17.5 

17.6 

—.1 

17.5 

17.3 

.2 

17.3 

17.3 

0 

4000 

15.8 

15.8 

0 

15.8 

15.5 

.3 

15.6 

15.5 

.1 

0 

39.7 

39.8 

—.1 

38.9 

38.8 

.1 

38.7 

38.7 

0 

500 

34.8 

34.8 

0 

34.1 

34.0 

.1 

34.0 

34.3 

—.3 

1000 

30.7 

30.7 

0 

30.1 

30.2 

— 1 

30.0 

29.9 

.1 

1500 

27.2 

27.2 

0 

26.7 

26.9 

2 

26.6 

26.4 

.2 

2000 

24.2 

24.2 

0 

23.8 

24.0 

—.2 

23.7 

23.5 

.2 

2500 

21.7 

21.8 

—.1 

21.3 

21.4 

—.1 

21.2 

21.1 

.1 

3000 

19.4 

19.4 

0 

19.1 

19.2 

—.1 

19.0 

18.9 

.1 

3500 

17.4 

17.4 

0 

17.2 

17.2 

0 

17.1 

16.8 

.3 

4000 

15.7 

15.6 

.1 

15.5 

15.4 

.1 

15.4 

15.2 

.2 

302 


1.  Experiments  with  the  Brass  Sphere  and  Long  Suspension.  —  Continued. 


m 

Computed 

ObserveJ 

C—0 

Computed 
An 

Observed 

C—0 

Computed 
An 

Observed 

C—0 

0 

38.7 

38.7 

0 

39.3 

39.3 

0 

39.1 

39.2 

—.1 

500 

34.0 

34.1 

—.1 

34.5 

34.7 

—.2 

34.3 

34.2 

.1 

1000 

30.0 

30.0 

0 

30.4 

30.2 

.2 

30.3 

30.3 

0 

1500 

2G.6 

26.6 

0 

27.0 

27.0 

0 

26.9 

27.0 

—.1 

2000 

23.7 

23.6 

.1 

24.0 

24.1 

—.1 

23.9 

23.9 

0 

2500 

21.2 

21.2 

0 

21.5 

21.5 

0 

21.4 

21.4 

0 

3000 

19.0 

19.0 

0 

19.3 

19.3 

0 

19.2 

19.3 

—.1 

3500 

17.1 

16.9 

.2 

17.3 

17.3 

0 

17.2 

17.2 

0 

4000 

15.4 

15.3 

.1 

15.6 

15.5 

.1 

15.5 

15.5 

0 

0 

39.0 

39.0 

0 

39.8 

39.7 

.1 

500 

34.2 

34.1 

.1 

34.9 

34.9 

0 

1000 

30.2 

30.1 

.1 

30.8 

30.8 

0 

1500 

26.8 

26.8 

0 

27.3 

27.2 

.1 

2000 

23.9 

23.7 

.2 

24.3 

24.3 

0 

2500 

21.3 

21.2 

0 

21.7 

21.7 

0 

3000 

19.1 

19.2 

—.1 

19.4 

19.4 

0 

3500 

17.2 

17.2 

0 

17.5 

17.4 

.1 

4000 

15.5 

15.4 

.1 

15.7 

15.6 

.1 

2. 

Experiments  with  the  Brass  Sphere  and  the  Short  Suspension 

m 

Comimted 
An 

Observed 
An 

C     0 

Computed 
An 

Observed 

C—0 

Computed 
An 

Observed 
Am 

C      0 

0 

14.4 

14.65 

—.2 

13.2 

13.5 

—.3 

12.4 

12.4 

0 

560 

13.5 

13.7 

—.2 

12.4 

12.7 

—.3 

11.7 

11.6 

.1 

1120 

12.7 

12.8 

—.1 

11.7 

11.9 

—.2 

11.0 

10.9 

.1 

1680 

12.0 

11.9 

,1 

11.0 

11.0 

.0 

10.3 

10.2 

.1 

2240 

11.3 

11.0 

.3 

10.4 

10.3 

.1 

9.7 

9.6 

.1 

2800 

10.6 

10.3 

.3 

9.7 

9.7 

0 

9.2 

9.0 

.2 

3360 

10.0 

9.6 

.4 

9.2 

9.0 

.2 

8.6 

8.5 

.1 

3920 

9.4 

8.9 

•i 

8.6 

8.4 

.2 

8.1 

8.0 

.1 

4480 

8.8 

8.3 

.f 

8.1 

7.9 

.2 

7.6 

7.5 

.1 

5040 

8.3 

7.8 

.5 

7.6 

7.4 

.2 

7.2 

7.1 

.1 

5600 

7.8 

7.3 

.5 

7.2 

7.0 

.2 

6.8 

6.7 

.1 

0 

12.2 

12.3 

—.1 

11.5 

11.6 

—.1 

12.2 

12.2 

0 

560 

11.5 

11.5 

0 

10.9 

10.9 

0 

11.5 

11.5 

0 

1120 

10.8 

10.8 

0 

10.3 

10.3 

0 

10.8 

10.9 

— 1 

1680 

10.2 

10.1 

.1 

9.7 

9.7 

0 

10.2 

10.3 

—.1 

2240 

9.6 

9.5 

.1 

9.1 

9.1 

0 

9.6 

9.7 

—.1 

2800 

9.0 

8.9 

.1 

8.6 

8.6 

0 

9.0 

9.1 

— 1 

3360 

8.5 

8.4 

.1 

8.1 

8.15 

—.1 

8.5 

8.5 

0 

3920 

8.0 

8.0 

0 

7.6 

7.7 

—.1 

8.0 

8.1 

—.1 

4480 

7.5 

7.5 

0 

7.1 

7.3 

.2 

7.5 

7.7 

—.2 

5040 

7.1 

7.0 

.1 

6.7 

6.9 

.2 

7.1 

7.2 

—.1 

5600 

6.7 

6.5 

.2 

6.3 

6.4 

—.2 

6.7 

6.8 

—.1 

303  — 


2.    Exjierlments  with  the  Brass  Sphere  and  the  Short  Suspensmt.  —  Continued. 


m 

Computed 

An 

Observe' I 
An 

0—0 

Computed 
A^ 

Observed 

0     0 

Computed 

Observed 
An 

0—0 

0 

12.5 

12.3 

.2 

12.8 

12.7 

.1 

12.9 

12.8 

.1 

560 

11.8 

11.7 

.1 

12.0 

11.95 

.1 

12.1 

12.0 

.1 

1120 

11.1 

11.0 

.1 

11.3 

11.3 

0 

11.4 

11.3 

.1 

1G80 

10.4 

10.4 

0 

10.7 

10.7 

0 

10.8 

10.7 

.1 

2240 

9.8 

9.8 

0 

10.0 

10.15 

—.1 

10.1 

10.2 

—.1 

2800 

9.2 

9.2 

0 

9.5 

9.5 

0 

9.5 

9.7 

/2 

3369 

8.7 

8.75 

—.1 

8.9 

8.9 

0 

9.0 

9.1 

—.1 

3920 

8.2 

8.3 

—.1 

8.4 

8.4 

0 

8.4 

8.6 

—.2 

4480 

7.7 

7.9 

—.2 

7.9 

8.0 

—.1 

7.9 

8.1 

—.2 

5040 

7.2 

7.45 

— .2 

7.4 

7.6 

—.2 

7.5 

7.7 

—.2 

5600 

6.8 

7.0 

—.2 

7.0 

7.2 

—.2 

7.0 

7.2 

,2 

0 

13.0 

12.9 

.1 

10.9 

10.9 

0 

13.4 

13.2 

.2 

560 

12.2 

12.1 

.1 

10.4 

10.3 

.1 

12.6 

12.5 

.1 

1120 

11.5 

11.4 

.1 

9.7 

9.7 

0 

11.9 

11.8 

.1 

1680 

10.8 

10.8 

0 

9.2 

9.2 

0 

1J.2 

11.2 

0 

2240 

10.2 

10.3 

—.1 

8.6 

8.7 

—.1 

10.5 

10.6 

—.1 

2800 

9.6 

9.8 

,2 

8.1 

8.2 

—.1 

9.9 

9.9 

0 

3360 

9.0 

9.2 

—.2 

7.6 

7.7 

—.1 

9.3 

9.3 

0 

3920 

8.5 

8.8 

—.3 

7.2 

7.2 

0 

8.8 

8.8 

0 

4480 

8.0 

8.2 

—.2 

6.8 

6.8 

0 

8.3 

8.3 

0 

5040 

7.5 

7.8 

—.3 

6.4 

6.4 

0 

7.8 

7.85 

—.1 

5600 

7.1 

7.4 

—.3 

6.0 

6.0 

0 

7.3 

7.45 

—.1 

0 

13.3 

13.3 

0 

11.1 

11.3 

-.2 

12.4 

12.5 

—.1 

560 

12.5 

12.5 

0 

10.5 

10.5 

0 

11.7 

11.7 

0 

1120 

11.8 

11.8 

0 

9.9 

9.8 

.1 

11.0 

10.9 

.1 

1680 

11.1 

11.1 

0 

9.3 

9.3 

0 

10.3 

10.2 

.1 

2240 

10.4 

10.5 

—.1 

8.8 

8.8 

0 

9.7 

9.6 

.1 

2800 

9.8 

9.8 

0 

8.3 

8.3 

0 

9.2 

9.0 

.2 

3360 

9.2 

9.2 

0 

7.8 

7.8 

0 

8.6 

8.5 

.1 

3920 

8.7 

8.7 

0 

7.3 

7.2 

.1 

8.1 

8.0 

.1 

4480 

8.2 

8.2 

0 

6.9 

6.8 

.1 

7.6 

7.6 

0 

5040 

7.7 

7.7 

0 

6.5 

6.4 

.1 

7.2 

7.2 

0 

5600 

7.3 

7.3 

0 

6.1 

6.1 

0 

6.8 

6.8 

0 

0 

11.7 

11.8 

—.1 

560 

11.1 

11.1 

0 

1120 

10.4 

10.4 

0 

1680 

9.8 

9.8 

0 

2240 

9.3 

9.2 

.1 

2800 

8.7 

8.7 

0 

3360 

8.2 

8.2 

0 

3920 

7.7 

7.7 

0 

4480 

7.3 

7.3 

0 

5040 

6.8 

6.8 

0 

5600 

6.4 

6.4 

0 

304 


3.  Experh 

nents  with  the  Ivory 

Sphere  and  Long  S 

Hspension. 

m 

Computed 

Obserred 

C     0 

Computed 

Observed 

C     0 

Computed 

Observed 

0—0 

0 

36.5 

36.4 

.1 

38.9 

38.9 

0 

38.7 

38.6 

.1 

500 

21.5 

22.0 

—.5 

22.7 

22.7 

0 

22.6 

22.7 

—.1 

1000 

13.5 

13.2 

.3 

14.2 

14.3 

—.1 

14.1 

14.3 

—.2 

0 

38.9 

38.9 

0 

37.9 

37.8 

.1 

37.9 

37.9 

0 

500 

22.7 

22.9 

,2 

22.2 

22.6 

—.4 

22.2 

22.4 

—.2 

1000 

14.2 

14.5 

—.3 

13.9 

14.3 

—.4 

13.9 

14.0 

—.1 

0 

39.1 

39.2 

—.1 

37.4 

37.5 

—.1 

38.5 

38.5 

0 

500 

22.7 

22.4 

.3 

21.9 

21.7 

.2 

22.5 

22.3 

.2 

1000 

14.2 

13.7 

.5 

13.7 

12.9 

.8 

14.0 

14.2 

—.2 

0 

38.4 

38.4 

0 

37.0 

37.1 

—.1 

37.3 

37.3 

0 

500 

22.4 

22.0 

.4 

21.7 

21.1 

.6 

21.9 

21.8 

.1 

1000 

14.0 

14.0 

0 

13.6 

13.4 

.2 

13.7 

13.9 

— 2 

0 

37.2 

37.3 

—.1 

36.8 

36.8 

0 

37.1 

36.9 

.2 

500 

21.8 

21.7 

.1 

21.6 

21.7 

—.1 

21.8 

22.1 

—.3 

1000 

13.7 

13.8 

—.1 

13.6 

13.4 

.2 

13.7 

13.9 

— 2 

0 

34.7 

34.7 

0 

500 

20.5 

20.6 

—.1 

1000 

13.3 

13.0 

.3 

1.  Experiments  with  the  Ivory 

Sphere  and  Short  S 

uspension. 

m 

Computed 

Observed 

0     0 

Computed 

Observed 

0     0 

Computed 
A,n 

Observed 

0      0 

0 

12.3 

12.3 

0 

13.6 

13.6 

0 

13.9 

14.0 

—.1 

650 

9.3 

9.3 

0 

10.1 

10.0 

.1 

10.3 

10.1 

.2 

1300 

7.1 

7.2 

—.1 

7.6 

7.8 

—.2 

7.8 

7.8 

0 

1950 

5.4 

5.7 

—.3 

5.8 

5.9 

—.1 

5.9 

5.8 

.1 

2600 

4.2 

4.3 

—.1 

4.4 

4.3 

.1 

4.5 

4.3 

.2 

0 

13.0 

13.1 

—.1 

14.8 

14.9 

—.1 

14.3 

14.3 

0 

650 

9.9 

9.9 

0 

10.9 

10.9 

0 

10.6 

10.7 

—.1 

1300 

7.5 

7.5 

0 

8.2 

8.0 

.2 

8.0 

8.0 

0 

1950 

5.7 

5.7 

0 

6.2 

6.0 

.2 

6.0 

6.0 

0 

2600 

4.5 

4.5 

0 

4.8 

4.5 

.3 

4.6 

4.6 

0 

0 

12.9 

13.1 

.2 

14.0 

14.0 

0 

13.2 

13.0 

.2 

650 

9.6 

9.6 

0 

10.4 

10.4 

0 

19.8 

19.9 

—.1 

1300 

7.2 

7.0 

.2 

7.8 

8.0 

—.2 

7.3 

7.4 

—.1 

1950 

5.5 

5.4 

.1 

5.9 

6.1 

—.2 

5.6 

5.9 

—.3 

2600 

4.2 

4.1 

.1 

4.5 

4.5 

0 

4.3 

4.4 

—.1 

0 

13.3 

13.1 

.2 

16.0 

16.0 

0 

16.8 

16.8 

0 

650 

9.8 

10.0 

—.2 

11.8 

11.8 

0 

12.4 

12.5 

—.1 

1300 

7.4 

7.3 

.1 

8.9 

8.8 

.1 

9.3 

9.4 

—.1 

1950 

5.6 

5.8 

—.2 

6.7 

6.8 

—.1 

7.1 

7.1 

0 

2600 

4.3 

4.5 

—.2 

5.2 

5.2 

0 

5.4 

5.5 

—.1 

305 


4.  Experiments  with  the  Ivory  Sphere  and  Short  Suspension.  —  Continued. 


in 

Computed 

An 

Observed 

C     0 

Computed 

Observed 

C     0 

Computed 
An 

Observed 
An 

C     0 

0 

16.6 

16.6 

0 

17.8 

18.0 

—.2 

16.3 

16.3 

0 

650 

12.2 

12.1 

.1 

13.0 

13.0 

0 

12.0 

12.2 

—.2 

1300 

9.2 

9.2 

0 

9.7 

9.5 

.2 

9.0 

9.1 

—.1 

rjuo 

7.0 

7.0 

0 

7.3 

7.1 

.2 

6.8 

7.0 

—.2 

2600 

5.3 

5.5 

.2 

5.6 

5.5 

.1 

5.2 

5.2 

0 

0 

16.1 

10.0 

.1 

650 

11.8 

12.0 

.2 

1300 

8.8 

9.0 

—.2 

1950 

6.7 

6.8 

—.1 

2600 

5.1 

5.1 

0 

5.  Experiments  with  the  Full  Cylinder  and  Long  Suspension. 


m 

Computed 

Observed 

^4,„ 

C     0 

Computed 
An 

Observed 

r  .0 

Computed 

Observed 

C     0 

0 

39.8 

39.8 

0 

41.5 

41.2 

.3 

38.3 

38.9 

—.6 

500 

35.8 

35.9 

—.1 

37.3 

37.3 

0 

84.5 

34.3 

.2 

1000 

32.4 

32.2 

.2 

33.6 

33.8 

—.2 

31.2 

30.8 

.4 

1500 

29.4 

29.4 

0 

30.5 

30.7 

—.2 

28.4 

27.8 

.6 

2000 

26.7 

26.6 

.1 

27.7 

27.8 

—.1 

25.8 

25.2 

.6 

2500 

24.3 

24.3 

0 

25.2 

25.3 

—.1 

23.6 

22.9 

.7 

3000 

26.3 

22.0 

.3 

23.0 

23.2 

—.2 

21.6 

20.6 

1.0 

3500 

20.4 

20.3 

.1 

21.1 

21.3 

—.2 

19.8 

18.7 

1.1 

4000 

18.7 

19.0 

— 3 

19.3 

19.5 

—.2 

18.2 

17.1 

1.1 

0 

39.4 

39.6 

2 

41.0 

41.5 

—.5 

41.7 

41.8 

—.1 

500 

35.5 

35.3 

.2 

36.9 

36.9 

0 

37.5 

37.6 

—.1 

1000 

32.1 

31.9 

.2 

33.3 

32.9 

.4 

33.8 

33.5 

.3 

1500 

29.1 

29.0 

.1 

30.1 

29.9 

.2 

30.6 

30.5 

.1 

2000 

26.5 

26.2 

.3 

27.4 

27.1 

.3 

27.8 

27.7 

.1 

2500 

24.1 

24.0 

.1 

25.0 

24.6 

.4 

25.3 

25.4 

—.1 

3000 

22.1 

22.0 

.1 

22.8 

22.5 

.3 

23.1 

23.2 

—.1 

3500 

20.2 

20.0 

.2 

20.9 

20.6 

.3 

21.2 

21.2 

0 

4000 

18.6 

18.3 

.3 

19.1 

19.0 

.1 

19.4 

19.3 

.1 

0 

39.5 

39.2 

.3 

40.2 

40.3 

—.1 

42.7 

42.7 

0 

500 

35.6 

35.6 

0 

36.2 

3G.1 

.1 

38.3 

38.1 

.2 

1000 

32.1 

32.4 

—.3 

32.7 

32.7 

0 

34.5 

34.6 

—.1 

1500 

29.2 

29.4 

.2 

29.6 

30.0 

—.4 

31.2 

31.4 

.2 

2000 

26.5 

26.6 

—.1 

26.9 

27.1 

—.2 

28.4 

28.3 

.1 

2500 

24.2 

24.3 

—.1 

24.5 

24.7 

—.2 

25.8 

25.9 

—.1 

3000 

22.1 

22.3 

.2 

22.4 

22.6 

—.2 

23.6 

23.6 

0 

3500 

20.3 

20.5 

—.2 

20.6 

207 

—.1 

21.6 

21.6 

0 

4000 

18.6 

18.7 

—.1 

18.9 

18.9 

0 

19.8 

19.9 

—.1 

39 


306  — 


5.  Experiments  with  the  Fall   Ctjlinder  and  Loncj  Suspension.  —  Continued. 


m 

Computed 

Observed 
An 

C—C 

i    Computed 
An 

Observed 

C—0 

Computed 
An 

Observed 

An 

C—0 

0 

42.3 

42.5 

—.2 

43.2 

43.1 

.1 

42.0 

41.8 

.2 

500 

38.0 

37.9 

.1 

38.8 

38.8 

0 

37.7 

38.4 

—.7 

1000 

34.2 

34.0 

.2 

34.9 

35.0 

—.1 

34.0 

34.0 

0 

1500 

31.0 

31.1 

—.1 

31.6 

31.6 

0 

30.8 

30.9 

—.1 

2000 

28.1 

28.1 

0 

28.7 

28.6 

.1 

28.0 

28.1 

—.1 

2500 

25.6 

25.5 

.1 

26.1 

26.1 

0 

25.5 

25.7 

—.2 

3000 

23.4 

23.5 

—.1 

23.8 

23.9 

—.1 

23.3 

23.5 

—.2 

3500 

21.4 

21.5 

—.1 

21.8 

21.7 

.1 

21.3 

21.5 

—.2 

4000 

19. G 

19.4 

.2 

20.0 

20.0 

0 

19.5 

19.6 

—.1 

0 

41.5 

41.4 

.1 

41.4 

41.2 

.2 

41.6 

41.4 

.2 

500 

37.3 

37.2 

.1 

37.2 

37.3 

—.1 

37.4 

37.4 

0 

1000 

33.6 

33.5 

.1 

33.6 

33.6 

0 

33.7 

33.8 

—.1 

1500 

30.5 

30.5 

0 

30.4 

30.4 

0 

30.5 

30.6 

—.1 

2000 

27.7 

27.9 

—.2 

27.6 

27.6 

0 

27.7 

27.9 

.2 

2500 

25.2 

25.3 

—.1 

25.2 

25.2 

0 

25.3 

25.5 

—.2 

3000 

23.0 

23.1 

—.1 

23.0 

23.1 

—.1 

23.1 

23.2 

—.1 

3500 

21.1 

21.3 

—.2 

21.0 

21.2 

—.2 

21.1 

21.3 

.2 

4000 

19.3 

19.4 

—.1 

19.3 

19.1 

.2 

19.5 

19.6 

—.1 

0 

41.4 

41.1 

.3 

40.5 

40.3 

.2 

39.3 

39.4 

—.1 

500 

37.2 

37.2 

0 

36.4 

36.5 

—.1 

35.4 

35.3 

.1 

1000 

33.6 

33.6 

0 

32.9 

33.0 

—.1 

32.0 

32.0 

0 

1500 

30.4 

30.5 

—.1 

29.8 

29.8 

0 

29,0 

29.0 

0 

2000 

27.6 

27.7 

—.1 

27.1 

27.0 

.1 

2G.4 

26.4 

0 

2500 

25.2 

25.2 

0 

24.7 

24.6 

.1 

24.1 

24.1 

0 

3000 

23.0 

23.1 

1 

22.6 

22.5 

.1 

22.0 

22.1 

—.1 

3500 

21.0 

21.2 

2 

20.7 

20.7 

0 

20.2 

20.3 

—.1 

4000 

19.3 

19.5 

2 

19.0 

19.0 

0 

18.5 

18.5 

0 

0 

38.0 

38.3 

— 3 

39.6 

39.5 

.1 

42.0 

41.8 

.2 

500 

34.1 

33.5 

.6 

35.6 

35.6 

0 

37.7 

37.9 

— .2 

1000 

30.9 

30.4 

.5 

32.2 

32.1 

.1 

34.0 

34.2 

,2 

1500 

28.0 

27.8 

.2 

29.2 

29.3 

— 1 

30.8 

31.0 

.2 

2000 

25.5 

25.3 

.2 

26.(i 

26.7 

— 1 

28.0 

28.2 

2 

2500 

23.2 

23.0 

.2 

24.2 

24.3 

— 1 

25.5 

25.6 

— 1 

3000 

21.3 

21.3 

0 

22.2 

22:3 

— 1 

23.3 

23.4 

— 1 

3500 

19.6 

19.6 

0 

20.3 

20.4 

— 1 

21.3 

21.5 

— 2 

4000 

18.0 

17.8 

.2 

18.6 

18.7 

— 1 

19.5 

19.6 

— 1 

0 

42.1 

42.0 

.1 

41.7 

41.7 

0 

40.6 

40.6 

0 

500 

37.8 

37.9 

—.1 

37.4 

37.4 

0 

36.5 

36.6 

—.1 

1000 

34.1 

34.1 

0 

33.8 

33.9 

—.1 

33.0 

33.0 

0 

1500 

30.9 

30.9 

0 

30.6 

30.6 

0 

29.9 

30.0 

— .1 

2000 

28.0 

28.1 

—.1 

27.8 

27.9 

—.1 

27.2 

27.2 

0 

2500 

25.5 

25.7 

—.2 

25.3 

25.5 

.2 

24.8 

24.8 

0 

3000 

23.3 

23.3 

0 

23.1 

23.2 

—.1 

22.6 

22.7 

— .1 

3500 

21.3 

21.3 

0 

21.2 

21.4 

—.2 

20.7 

20.8 

— .1 

4000 

19.5 

19.5 

0 

19.4 

19.6 

—.2 

19.0 

18.9 

.1 

307  — 


6.  Experiments  rvith  the  Full  Cjjlinder  and  Short  Suspension. 


m 

Computed 

Obseryed 
An 

C       0 

Computed 
An 

Observed 
An 

C       0 

Computed 
An 

Observed 
An 

C—0 

0 

12.4 

12.45 

0 

12.1 

12.15 

0 

13.2 

13.15 

0 

730 

11.7 

11.6 

.1 

11.4 

11.35 

0 

12.5 

12.5 

0 

1400 

11.0 

11.0 

0 

10.8 

10.65 

.1 

11.8 

11.95 

— .1 

2100 

10.4 

10.4 

0 

10.2 

10.25 

—.1 

11.2 

11.1 

2920 

9.9 

9.8 

.1 

9.6 

9.55 

.1 

10.5 

10.55 

— .1 

3650 

9.3 

9.3 

0 

9.1 

9.1 

0 

9.9 

10.0 

— ,1 

4380 

8.8 

8.85 

0 

8.6 

8.6 

0 

9.4 

9.45 

— .1 

5110 

8.3 

8.35 

0 

8.1 

8.15 

0 

8.9 

8.9 

0 

5840 

7.9 

7.85 

0 

7.7 

7.65 

0 

8.4 

8.5 

—.1 

0 

13.0 

13.0 

0 

12.1 

12.05 

0 

12.6 

12.55 

0 

730 

12.3 

12.25 

0 

11.5 

11.7 

—.2 

1 2.0 

12.0 

0 

1460 

11.6 

11.55 

0 

10.9 

11.05 

—.1 

11.3 

11.3 

0 

2190 

10.9 

11.05 

—.1 

10.4 

10.5 

—.1 

10.8 

10.75 

0 

2920 

10.3 

10.3 

0 

9.8 

9.95 

—.1 

10.2 

10.1 

.1 

3650 

9.8 

9.75 

0 

9.5 

9.45 

.1 

9.7 

9.65 

.1 

4380 

9.2 

9.2 

0 

8.9 

9.05 

—.1 

9.2 

9.25 

0 

5110 

8.7 

8.8 

—.1 

8.5 

8.6 

—.1 

8.8 

8.8 

0 

5840 

8.2 

8.2 

0 

8.0 

8.05 

0 

8.4 

8.35 

0 

0 

12.4 

12.4 

0 

12.4 

12.4 

0 

13.5 

13.5 

0 

730 

11.8 

11.65 

.1 

11.8 

11.75 

0 

12.8 

12.75 

.1 

1460 

11.2 

11.15 

.1 

11.2 

11.2 

0 

12.2 

12.35 

—.1 

2190 

10.6 

10.55 

.1 

10.6 

10.6 

0 

11.6 

11.65 

0 

2920 

10.1 

10.2 

— 1 

10.1 

10.05 

.1 

11.0 

11.0 

0 

3650 

9.6 

9.6 

0 

9.6 

9.55 

.1 

10.5 

10.55 

0 

4380 

9.2 

9.15 

0 

9.2 

9.15 

.1 

10.0 

10.0 

0 

5110 

8.7 

8.7 

0 

8.7 

8.65 

.1 

9.5 

9.55 

0 

5840 

8.3 

8.35 

0 

8.4 

8.3 

.1 

9.1 

9.1 

0 

0 

13.0 

12.95 

0 

13.6 

13.6 

0 

13.9 

14.0 

—.1 

730 

12.4 

12.4 

0 

12.9 

13.0 

—.1 

13.2 

13.2 

0 

1460 

11.7 

11.85 

— 1 

12.3 

12.05 

.2 

12.5 

12.65 

—.1 

2190 

11.2 

11.3 

— 1 

11.6 

11.5 

.1 

11.9 

11.95 

0 

2920 

10.6 

10.8 

2 

11.1 

11.0 

.1 

11.3 

11.4 

—.1 

3650 

10.1 

10.1 

0 

10.5 

10.55 

0 

10.7 

10.55 

.2 

4380 

9.7 

9.7 

0 

10.0 

10.0 

0 

10.2 

10.25 

0 

5110 

9.2 

9.2 

0 

9.5 

9.55 

0 

9.7 

9.8 

—.1 

5840 

8.8 

8.9 

—.1 

9.0 

9.0 

0 

9.3 

9.2 

.1 

—  308 


7. 

Experiments  with  the  Emptij   Cijlinder  and  Long 

Suspension 

m 

Computed 

Observed 

C     0 

Computed 
An 

Observed 

C     0 

Computed 

Observed 

C       0 

0 

37.5 

37.7 

—.2 

38.2 

37.8 

.4 

37.6 

37.6 

0 

500 

28.1 

27.7 

.4 

28.6 

28.8 

.2 

28.2 

28.3 

.1 

1000 

21.4 

21.0 

.4 

21.8 

22.0 

—.2 

21.5 

21.4 

1500 

16.5 

16.6 

—.1 

16.8 

16.7 

.1 

16.6 

16.7 

— .1 

2000 

12.9 

13.1 

— .2 

13.1 

13.0 

.1 

12.9 

13.0 

—.1 

0 

38.0 

37.9 

.1 

39.2 

38.8 

.4 

38.9 

38.8 

500 

28.4 

28.6 

—.2 

29.3 

29.4 

—.1 

29.1 

29.2 

— ,1 

1000 

21.7 

21.6 

.1 

22.3 

22.4 

—.1 

22.1 

22.3 

— .2 

1500 

1G.7 

16.8 

—.1 

17.2 

17.2 

0 

17.1 

17.2 

•  •'■ 

2000 

13.0 

13.0 

0 

13.4 

13.4 

0 

13.3 

13.2 

0 

40.3 

40.4 

—.1 

40.8 

40.8 

0 

38.0 

38.0 

0 

500 

30.0 

29.9 

.1 

30.4 

30.4 

0 

28.4 

28.4 

0 

1000 

22.8 

22.5 

.3 

23.1 

23.1 

0 

21.7 

21.7 

0 

1500 

17.6 

17.8 

—.2 

17.7 

17.9 

—.2 

16.7 

16.8 

—.1 

2000 

13.7 

13.9 

—.2 

13.8 

13.9 

—.1 

13.0 

13.0 

0 

0 

37.9 

37.9 

0 

40.2 

40.2 

0 

39.9 

40.0 

—.1 

500 

28.4 

28.4 

0 

30.0 

30.0 

0 

29.7 

29.6 

.1 

1000 

21.6 

21.5 

.1 

22.8 

22.9 

—.1 

22.6 

22.5 

.1 

1500 

16.7 

16.8 

—.1 

17.5 

17.6 

—.1 

17.4 

17.3 

.1 

2000 

13.0 

13.0 

0 

13.6 

13.9 

—.3 

13.6 

13.6 

0 

0 

39.8 

40.0 

2 

39.2 

39.1 

.1 

40.7 

40.5 

.2 

500 

29.7 

29.7 

0 

29.3 

29.6 

—.3 

30.3 

30.4 

—.1 

1000 

22.6 

22.4 

.2 

22.3 

22.3 

0 

23.0 

23.1 

—.1 

2500 

17.4 

16.7 

.7 

17.2 

17.2 

0 

17.7 

17.6 

.1 

2000 

13.5 

13.2 

.3 

13.4 

13.4 

0 

13.8 

13.7 

.1 

0 

40.4 

40.4 

0 

40.1 

40.2 

—.1. 

40.4 

40.4 

0 

500 

30.1 

30.3 

—.2 

29.9 

29.9 

0 

30.1 

30.2 

—.1 

1000 

22.9 

22.7 

.2 

22.7 

22.7 

0 

22.9 

22.7 

.2 

1500 

17.6 

17.5 

.1    . 

17.5 

17.5 

0 

17.6 

17.6 

•0 

2000 

13.7 

13.7 

0 

13.6 

13.6 

0 

13.7 

13.6 

.1 

0 

38.9 

38.9 

0 

38.6 

38.8 

2 

38.7 

38.7 

0 

500 

29.1 

29.1 

0 

28.8 

28.6 

.2 

28.9 

28.8 

.1 

1000 

22.1 

21.9 

.2 

22.0 

21.9 

.1 

22.0 

22.2 

— .2 

1500 

17.0 

17.0 

0 

16.9 

16.9 

0 

17.0 

17.1 

—.1 

2000 

13.3 

13.4 

— 1 

13.2 

13.2 

0 

13.2 

13.4 

.2 

0 

39.5 

39.5 

0 

38.1 

38.0 

.1 

38.5 

38.5 

0 

500 

29.5 

29.8 

—.3 

28.5 

28.6 

—.1 

28.8 

29.0 

— 2 

1000 

22.4 

22.4 

0 

21.7 

21.8 

—.1 

21.9 

21.9 

0 

1500 

17.3 

17.2 

.1 

16.8 

16.6 

.2 

16.9 

16.8 

.1 

2000 

13.4 

13.4 

0 

13.0 

12.9 

.1 

13.2 

13.0 

.2 

—  309  — 


7.    Experiments  with  the  Empty   Cylinder  and  Long  Suspension.  —  Continued. 


m 

Computed 

Observed 

C     0 

Computed 
An 

Observed 
An 

C—0 

Computed 

Observed 

An 

C—0 

0 

39.2 

38.8 

A 

38.5 

38.4 

.1 

39.0 

39.0 

0 

500 

29.3 

29.5 

—.2 

28.8 

29.0 

—.2 

29.1 

29.2 

—.1 

1000 

22.3 

22.4 

—.1 

21.9 

21.9 

0 

22.2 

22.2 

0 

1500 

17.2 

17.2 

0 

16.9 

16.9 

0 

17.1 

17.2 

—.1 

2000 

13.4 

13.6 

.2 

13.2 

18.2 

0 

13.3 

13.4 

—.1 

0 

39.9 

39.9 

0 

38.4 

38.4 

0 

38.9 

38.9 

0 

500 

29.8 

29.8 

0 

28.7 

28.8 

— 1 

29.1 

29.2 

—.1 

1000 

22.6 

22.8 

— .2 

21.9 

22.0 

1 

22.1 

22.1 

0 

1500 

17.4 

17.6 

— .2 

16.9 

17.0 

_.l 

17.1 

17.1 

0 

2000 

13.6 

13.6 

0 

13.1 

13.1 

0 

13.3 

13.2 

.1 

0 

38.6 

38.6 

0 

39.1 

39.1 

0 

36.8 

36.8 

0 

500 

28.9 

28.9 

0 

29.2 

29.3 

1 

27.6 

27.6 

0 

1000 

22.0 

21.9 

.1 

22.2 

22.1 

.1 

21.1 

21.1 

0 

1500 

16.9 

16.8 

.1 

17.1 

17.0 

.1 

16.3 

16.3 

0 

2000 

13.2 

13.1 

.1 

13.3 

13.1 

.2 

12.6 

12.6 

0 

0 

36.3 

36.3 

0 

38.3 

38.3 

0 

39.4 

39.3 

.1 

500 

27.3 

27.4 

1 

28.6 

28.7 

1 

29.4 

29.5 

—.1 

1000 

20.8 

21.0 

.2 

21.8 

21.8 

0 

22.4 

22.4 

0 

1500 

16.1 

16.1 

0 

16.8 

16.8 

0 

17.2 

17.3 

—.1 

2000 

12.5 

12.5 

0 

13.1 

13.1 

0 

13.4 

13.5 

—.1 

0 

40.1 

40.1 

0 

39.4 

39.3 

.1 

38.8 

38.8 

0 

500 

29.9 

29.9 

0 

29.4 

29.5 

— 1 

29.0 

29.4 

—.4 

1000 

22.7 

22.6 

.1 

22.4 

22.4 

0 

22.1 

22.0 

.1 

1500 

17.5 

17.4 

.1 

17.2 

17.3 

— 1 

17.0 

17.0 

0 

2000 

13.6 

13.5 

.1 

13.4 

13.5 

— 1 

13.2 

13.3 

—.1 

0 

40.0 

40.0 

0 

38.7 

38.7 

0 

38.8 

38.9 

—.1 

500 

29.8 

30.0 

/2 

28.9 

28.9 

0 

29.0 

28.9 

.1 

1000 

22.7 

22.5 

.2 

22.0 

21.9 

.1 

22.1 

22.0 

.1 

1500 

17.4 

17.3 

.1 

17.0 

16.9 

.1 

17.0 

17.1 

—.1 

2000 

13.6 

13.6 

0 

13.2 

13.2 

0 

13.2 

13.3 

—.1 

0 

39.1 

39.1 

0 

38.5 

38.5 

0 

38.7 

38.7 

0 

500 

29.2 

29.1 

.1 

28.8 

28.8 

0 

28.9 

29.0 

— 1 

1000 

22.2 

22.1 

.1 

21.9 

21.8 

.1 

22.0 

22.0 

0 

1500 

17.1 

17.1 

0 

16.9 

16.9 

0 

17.0 

16.9 

.1 

2000 

13.3 

13.3 

0 

13.2 

13.1 

.1 

13.2 

13.2 

0 

0 

37.7 

37.7 

0 

38.8 

38.8 

0 

38.1 

38.2 

—.1 

500 

28.2 

28.0 

.2 

29.0 

29.0 

0 

28.5 

28.5 

0 

1000 

21.5 

21.5 

0 

22.1 

22.0 

.1 

21.7 

21.5 

.2 

1500 

16.6 

16.7 

—.1 

17.0 

16.9 

.1 

16.8 

16.6 

2 

2000 

12.9 

12.9 

0 

13.2 

13.1 

.1 

13.0 

12.9 

.1 

—  310 


8.  Experiment^: 

ivith  the  Empti/   Cijlinder  and  the  Shm-t' Suspension. 

m 

Computed 
An 

Observed 

C—0 

Computed 

Observed 

^          /-^    Computed 

Observed   ' 
Aa 

C—0 

0 

11.4 

11.4 

0 

12.2 

12.3 

— 1 

13.3 

13.3 

0 

800 

9.0 

9.4 

.1 

10.1 

10.15 

0 

11.0 

10.95 

0 

IGOO 

7.9 

7.7 

2 

8.5 

8.45 

0 

9.1 

9.05 

0 

2400 

6.6 

6.5 

1 

7.1 

7.05 

0 

7.6 

7.55 

0 

3200 

5.5 

0.5 

.0 

6.0 

5.9 

.1 

6.3 

6.3 

0 

4000 

4.7 

4.6 

.1 

5.0 

4.9 

.1 

5.3 

5.3 

0 

4800 

4.0 

3.9 

.1 

4.2 

4.1 

.1 

4.4 

4.5 

—.1 

0 

13.3 

13.3 

0 

13.4 

13.4 

0 

12.1 

12.1 

0 

800 

11.0 

11.1 

—.1 

11.3 

11.3 

0 

10.2 

10.25 

0 

IGOO 

9.1 

9.15 

0 

9.5 

9.4 

.1 

8.7 

8.65 

0 

2400 

7.6 

7.8 

—.2 

8.0 

8.05 

0 

7.3 

7.4 

—.1 

3200 

6.3 

6.2 

.1 

6.8 

6.7 

.1 

6.2 

6.25 

0 

4000 

5.3 

5.4 

—.1 

5.8 

5.7 

.1 

5.3 

5.3 

0 

4800 

4.4 

4.45 

0 

4.9 

4.9 

0 

4.5 

4.5 

0 

0 

12.3 

12.2 

.1 

13.0 

13.0 

0 

13.2 

13.15 

0 

800 

10.5 

10.5 

0 

11.0 

11.0 

0 

11.2 

11.15 

0 

1000 

8.9 

9.0 

—.1 

9.4 

9.3 

.1 

9.5 

9.6 

— 1 

2400 

7.6 

7.85 

2 

8.0 

8.0 

0 

8.1 

8.0 

.1 

3200 

6.5 

6.7 

,2 

6.9 

6.9 

0 

6.9 

6.85 

0 

4000 

5.6 

5.75 

,2 

5.9 

5.9 

0 

5.9 

5.95 

0  . 

4800 

4.8 

4.95 

—.1 

5.0 

5.1 

—.1 

5.1 

5.1 

0 

0 

13.0 

12.95 

0 

14.1 

14.25 

—.1 

12.9 

12.9 

0 

800 

11.0 

11.05 

0 

11.9 

11.9 

0 

10.9 

10.95 

0 

IGOO 

9.4 

9.4 

0 

10.1 

10.1 

0 

9.3 

9.15 

.1 

2400 

8.0 

8.05 

—.1 

8.6 

8.65 

—.1 

7.9 

7.85 

0 

3200 

6.8 

6.9 

—.1 

7.3 

7.35 

—.1 

6.7 

6.85 

—.1 

4000 

5.8 

5.85 

0           6.2 

5.95 

.3 

5.7 

5.85 

—.1 

4800 

5.0 

5.0 

0           5.3 

5.25 

.1 

4.9 

4.9 

0 

In  the  computation  of  these  values,  there  has  been  no  regard 
to  the  resistance  arising  from  the  wires  of  suspension.  The  dif- 
ference between  the  vahies  of  ^  may  be  attributed  to  the  uncer- 
tainty of  the  observations,  and  those  of  H^  niay,  perhaps,  be  ac- 
counted for,  in  the  same  way.  The  vahie  of  H^  is  nearly  ten 
times  as  great  as  that  Avliich  is  given  by  the  observations  of  Borda 
upon  the  resistance  of  the  atmosphere.  It  must,  therefore,  be 
doubtful,  whether  the  observed  diminution  of  the  arcs  of  vibration 


—  311  — 

of  the  pendulum  is,  wholly  or  principally,  clue  to  the  medium  in 
which  it  vibrates,  or  to  some  more  latent  cause.  This  doubt  is 
much  increased  by  the  discussion  of  the  observations  of  Baily. 

534.  In  Baily's  experiments,  various  pendulums,  which  were 
mostly  spheres  and  cylinders,  were  vibrated  in  the  receiver  of  an 
air-jDump,  with  the  air  either  at  its  ordinary  pressure,  or  at  the 
small  density  of  about  one  thirtieth  of  an  atmosphere.  For  the 
full  and  exact  description  of  the  pendulums  the  original  memoir 
must  be  consulted,  but  the  following  brief  description  is  sufficient 
for  the  present  purpose.  Numbers  1,  2,  3,  and  4  are  spheres  of 
platina,  lead,  brass,  and  ivory,  all  of  the  same  diameter,  which  is 
somewhat  less  than  1 J  inches,  and  of  which  the  weights  with  their 
vibrating  appendages  are,  respectively  9050,  4648,  3217,  and  776 1 
grains.  Nos.  5,  6,  and  7  are  spheres  of  lead,  brass,  and  ivory,  all 
of  the  same  diameter,  which  is  2.06  inches,  and  of  which  the 
weights  are  respectively,  13019,  9302,  and  2066^  grains.  Nos.  8 
and  9  are  the  same  spheres  of  lead  and  ivory  with  those  of  Nos.  5 
and  7,  but  suspended  from  a  wire  passing  over  a  small  cylinder 
instead  of  from  a  knife  edge.  In  Nos.  10,  11,  12,  and  13  the 
vibrating  mass  was  a  brass  cylinder,  of  which  the  diameter  of  the 
base  is  2.06  inches,  the  altitude  2.06  inches,  and  the  weight  14190 
grains  ;  in  Nos.  10  and  13  the  axis  of  the  cylinder  coincides  with 
that  of  the  pendulum  rod,  but  the  rod  of  No.  13,  which  was  also 
adopted  in  Nos.  11  and  12,  w^as  a  thick  brass  wire  0.185  inch  in 
diameter,  37^  inches  long,  and  weighing  2050  grains  5  in  Nos.  11 
and  12  the  axis  of  the  cylinder  was  horizontal,  in  No.  11  it  was 
perpendicular  to  the  plane  of  vibration,  and  in  No.  12  it  was  in  the 
plane  of  vibration.  No.  14  is  a  cylinder  of  lead,  of  which  the 
diameter  of  the  base  is  2.06  inches,  the  altitude  4  inches,  the  weight 
34500  grains,  and  the  axis  coincident  w4th  the  rod  of  the  pendulum. 
In  Nos.  15,  16,  17,  18,  and  19  the  vibrating  mass  was  a  hollow  cyl- 


—  312  — 

inder  of  the  same  position  and  external  dimensions  with  No.  14 ; 
in  No.  15  both  ends  were  open ;  in  No.  16  the  top  was  open  and 
the  bottom  closed;  in  No.  17  the  top  was  closed  and  the  bottom 
open;  in  No.  18  both  ends  were  closed ;  in  No.  19  an  inner  sliding 
tube  was  removed  so  as  to  reduce  the  weight;  and  the  weights, 
with  the  inclosed  air,  were,  respectively,  8497,  8922,  8622,  9048, 
and  7250  grains.  No.  20  is  a  lens  of  lead  2.06  inches  in  diameter, 
an  inch  thick  in  the  middle,  with  a  flat  circumference  of  about  a 
quarter  of  an  inch  wide,  and  a  weight  of  6505  grains.  No.  21  is  a 
solid  copper  cylindrical  rod  of  0.41  inch  in  diameter,  58.8  inches 
long,  and  weighing  16810  grains.  In  Nos.  25,  26,  27,  28,  29,  30,  31, 
32,  33,  and  34,  the  vibrating  masses  were  convertible  pendulums, 
formed  of  plane  bars,  and  they  are  vibrated  successively  with  each 
of  their  points  of  suspension,  which  were  knife  edges;  in  Nos.  25 
and  26  the  bar  was  brass,  two  inches  wide,  three  eighths  of  an  inch 
thick,  62.2  inches  long,  and  weighing  121406  grains;  in  Nos.  27 
and  28  it  was  copper  of  the  same  width  with  the  brass  bar,  half 
an  inch  thick,  62.5  inches  long,  and  weighed  155750  grains;  in 
Nos.  29  and  30,  it  was  iron  of  the  same  width  and  thickness  with  the 
copper  bar,  62.1  inches  long,  and  weighed  140547  grains;  in  Nos.  31, 
32,  33,  and  34  it  was  a  doubly  convertible  brass  bar,  three  quarters  of 
an  inch  thick,  62  inches  long,  and  weighed  231437  grains.  In  Nos.  35, 
36,  37,  and  38,  a  doubly  convertible  pendulum,  made  of  a  brass  cylin- 
drical tube  of  Ij  inches  in  diameter,  56  inches  long,  and  weighing 
81047  grains  was  vibrated  upon  a  knife  edge  with  all  four  of  its 
planes  of  suspension.  No.  39  is  a  mercurial  pendulum.  Nos.  40  and 
41  are  clock  pendulums  in  which  the  vibrating  mass  was  a  leaden 
cylinder  1.8  inches  in  diameter,  13.5  inches  long,  and  weighing 
93844  grains ;  in  No.  40  it  was  suspended  from  a  spring,  by  a  cylin- 
drical rod  of  deal  of  three  eighths  of  an  inch  in  diameter,  and  in 
No.  41  by  a  flat  rod  of  deal  one    inch   wide,  0.14  inch  thick  in  the 


—  313  — 

middle  of  its  width  and  bevelled  on  each  side  to  a  thin  edge,  which 
was  opposed  to  the  direction  of  its  motion. 

In  the  discussion  of  Baily's  experiments,  the  value  of  II2  is 
neglected,  because  it  is  of  small  influence,  and  the  arcs  of  vibration, 
being  usually  given  only  for  the  beginning  and  end  of  the  experi- 
ment, are  just  sufficient  to  determine  one  of  the  quantities  H^ 
and  H2 ',  and  the  values  of  H^  are  not  reduced  to  the  same  density 
of  air.  The  ratio  of  the  value  of  Hi  for  the  ordinary  state  of  the 
air  to  its  value  in  the  exhausted  receiver,  varies  from  1.9  to  4.2,  in- 
stead of  being  about  30,  which  it  should  be  if  it  were  proportional 
to  the  density  of  the  air ;  the  value  of  this  ratio  in  the  following 
table  is  expressed  by  J.  The  total  resistance  to  the  motion  of  the 
pendulum,  supposed  to  be  proportional  to  the  velocity  is,  for  the 
imit  of  velocity,  expressed  by  H['  in  the  table ;  and  this  same  re- 
sistance, reduced  to  the  unit  of  Aveight,  is  expressed  by  Hi. 

The  observation  of  the  arcs  of  vibration  in  Baily's  experiments 
is  limited  to  the  initial  and  final  arcs,  and  the  direct  comparison  of 
the  computed  and  observed  arcs  is,  consequently,  quite  unnecessary, 
and  cannot  contribute  to  verify  the  accuracy  of  the  hypothesis  upon 
which  the  computation  is  based.  The  only  two  cases  in  which  an 
intermediate  arc  was  observed  with  Nos.  6  and  14  seem  to  sustain 
the  hypothesis  ;  for  they  differ  from  it  slightly,  but  in  oj^posite  direc- 
tions. 

The  diversity  of  the  values  of  H^  indicates  that  the  resisting 
force  of  the  motion  to  the  pendulum  demands  a  new  experimental 
investigation,  conducted  with  a  direct  object  to  its  determination ; 
and  that,  until  such  an  investigation  has  been  made,  the  length  of 
the  seconds  pendulum  must  be  regarded  as  liable  to  an  unknown 
error. 

40 


314 


Values  of  Hi  in  Bailij's  Experiments  upon  the   Vibrations  of  Pendulums. 


No.  of 
PenJulums. 

Barometer. 

H, 

Hi 

m' 

J 

1 

0.7089 

.0673 

.000077 

.000132 

2.68 

3 

0.7646 

.0702 

.000080 

.000384 

2.62 

2 

0.7523 

.0662 

.000075 

.000250 

2.55 

4 

0.7660 

.0561 

.000063 

.001272 

2.71 

6 

0.7638 

.0570 

.000123 

.000204 

2.74 

7 

0.7630 

.0538 

.000116 

.000864 

2.62 

5 

0.7644 

.0627 

.000128 

.000161 

3.18 

9 

0.7(;82 

.0589 

.000127 

.000945 

2.86 

8 

0.7677 

.1021 

.000219 

.000261 

2.92 

10 

0.7652 

.0651 

.000179 

.000194 

3.42 

11 

0.7637 

.0558 

.000270 

.000256 

2.62 

12 

0.7623 

.0603 

.000290 

.000277 

3.33 

13 

0.7552 

.0571 

.000235 

.000262 

2.98 

18 

0.7491 

.0535 

.000285 

.000484 

3.27 

15 

0.7554 

.0658 

.000350 

.000635 

4.10 

16 

0.7495 

.0595 

.000292 

.000505 

2.95 

17 

0.7584 

.0558 

.000297 

.000531 

3.39 

14 

0.7747 

.0592 

.000315 

.000140 

4.22 

19 

0.7620 

.0510 

.000272 

.000578 

3.33 

20 

0.7620 

.0656 

.000065 

.000156 

2.09 

21 

0.7575 

.0661 

.000742 

.000682 

2.72 

25 

0.7522 

.0789 

.005606 

.000333 

3.32 

2G 

0.7465 

.0756 

.004782 

.000319 

3.74 

31 

0.7522 

.1555 

.003666 

.000245 

3.32 

32 

0.7520 

.1581 

.003673 

.000245 

3.55 

34 

0.7529 

.1661 

.003772 

.000251 

3.72 

33 

0.7535 

.1417 

.003480 

.000232 

3.13 

35 

0.7595 

.0739 

.003091 

.000589 

3.48 

36 

0.7627 

.0660 

.002763 

.000526 

3.31 

37 

0.7577 

.0701 

.002931 

.000558 

3.39 

38 

0.7564 

.0659 

.002760 

.000526 

2.97 

39 

0.7622 

.001396 

.000209 

1.87 

41 

0.7573 

.0664 

.001260 

.000207 

2.52 

40 

0.7589 

.0769 

.001299 

.000213 

2.39 

—  315  — 


Values  of  III  '"  Bailij's  Exper'nnonts  upon  the   Vibrations  of  Pendulums.  —  Continued. 


Xo.  of 
Pendulums. 

Barometer. 

H, 

m 

Hi' 

J 

1 

0.0288 

.0251 

.000028 

.000049 

2.68 

3 

0.0294 

.0267 

.000031 

.000146 

2.62 

2 

0.02  Go 

.0259 

.000030 

.000098 

'2Jro 

4 

0.0347 

.0284 

.000024 

.000470 

2.71 

6 

0.0268 

.0285 

.000044 

.000074 

2.74 

7 

0.0270 

.0282 

.000044 

.000330 

2.62 

5 

0.0290 

.0275 

.000042 

.000050 

3.18 

9 

0.03  GO 

.028.2 

.000044 

.000331 

2.86 

8 

0.0299 

.0348 

.000075 

.000089 

2.92 

10 

0.0239 

.0190 

.000052 

.000057 

3.42 

11 

0.0478 

.0213 

.000103 

.000098 

2.62 

12 

0.0348 

.0182 

.000088 

.000083 

3.33 

13 

0.0370 

.0192 

.000092 

.000O.S9 

2.98 

18 

0.0300 

.0164 

.000087 

.000148 

3.27 

15 

0.0271 

.0164 

.000097 

.000148 

4.10  - 

16 

0.02  66 

.0186 

.000099 

.000171 

2.95 

17 

0.03  G2 

.0165 

.000088 

.000157 

3.39 

14 

0.0298 

.0139 

.000074 

.000033 

4.22 

19 

0.0305 

.0154 

.000083 

.000174 

3.33 

20 

0.0305 

.0313 

.000031 

.000074 

2.09 

21 

0.0288 

.0244 

.000274 

.000251 

2.72 

25 

0.0313 

.0238 

.001505 

.000101 

3.32 

26 

0.0325 

.0202 

.001277 

.000086 

3.74   . 

31 

0.0414 

.0469 

.001105 

.000074 

3.32 

32 

0.0391 

.0439 

.001034 

.000069 

3.55 

34 

0.0410 

.0431 

.001014 

.000067 

3.72 

33 

0.0463 

.0472 

.001111 

.000074 

3.13 

35 

0.0384 

.0213 

.000888 

.000170 

3.48 

36 

0.0367 

.0200 

.000834 

.000160 

3.31 

37 

0.0422 

.0206 

.000859 

.000166 

3.39 

38 

0.0412 

.0222 

.000930 

.000178 

2.97 

39 

0.0477 

.000747 

.000112 

1.87 

41 

0.0457 

.0263 

.000498 

.000083 

2.52 

40 

0.0434 

.0320 

.000543 

.000089 

2.39 

—  31G 


THE  TAUTOCHRONE. 


535.  The  consideration  of  the  pendulum  leads,  directly,  to 
the  investigation  of  that  curve,  upon  which  the  duration  of  the 
vibration  is  independent  of  the  length  of  the  arc  of  oscillation. 
Such  a  curve  is  called  a  iaidochrone,  and  is  readily  determined  when 
the  body  is  only  subject  to  the  action  of  fixed  forces. 

536.  If  the  force  which  acts  in  the  direction  of  the  motion 
of  the  body  is  denoted  by  S,  the  equation  of  its  motion  is 

In  the  case  in  which  /S'  is  a  function  of  s,  let  Sq  denote  the 
point,  at  which  the  velocity  vanishes,  or  the  extremity  of  the  arc 
of  vibration.     Hence 


v'  =  2Jy=.2{n-ll,)- 


and  if  the  origin  of  coordinates  is  at  the  point  of  maximum  velocity, 
the  time  of  vibration  is  determined  by  the  equation 


If  h  —  - 


Sn 


if  i2  is  a  function  of  s  expressed  by  X2,,  and  if  s  is  written  instead 
of  ^0,  the  value  of  T  becomes 

0 

In  order  that  the  special  value  of  the  arc  may  disappear  from 


—  317  — 
this  integral,  it  is  obvious  that  S2,  has  the  form 

which  reduces  the  value  of  T  to 

Jhs/Bs/il  —  fr)         \2B 

The  tangential  force  along  the  curve  is,  therefore, 

537.  If  ^  denotes  the  actual  force,  which  acts  upon  the  body 
in  the  direction  of/,  the  preceding  equation  gives  for  an  equation  of 
the  taiitochrone 

Fco8{^  —  2Bs=FI)J, 
or 

A—Bs^=zJf. 

538.  In  the  case  in  ivhich  the  hody  is  restricted  to  move  ttpon  a 
curve  ivhich  rotates  uniformly  about  a  fixed  axis,  the  equations  and 
notation  of  §468  combined  with  the  previous  section,  give  for 
the  equation  of  the  tautochrone 

A  —  Bs''=la''u'', 

which  may  assume  the  form 


S'    ,     u 


-4--  — 1 

in  which  a  and  b  are  constants. 

539.      When  the   revolving  curve  is  a  plane  curve,  and  situated  in 
the  same  plane  with  the  axis  of  revolution,  the  notation 

b  =:■  a  cot  i, 

s  :=:z  a  sin  ^=^  a  sin  (p  sin  /, 


—  318  — 

and  that  of  elliptic  functions  give 

u^h  cos ^ , 
z  =  -. — :  ^i  (f  —  b  cos  i  73 i  (p 


and  if  t  is  the  inclination  of  the  curve  to  the  axis  of  rotation,  its 
value  is 

sin  T  =  —  cot  i  tan  6 . 
The  maxiniuni  of  ii  is  b,  but  its  least  value,  corresponding  to 

or  ^  =  Ijl  i, 

is  11  =  b  cos  i ; 

and  the  corresponding  value  of  s  is 

s=:ljlasin/. 

The  curve  consists  of  several  branches,  which  form  cusps  by  their 
mutual  contact  at  their  extremities,  and  it  resembles  the  cycloid  in  its  general 
character. 

540.  In  the  case  of  a  heavy  body  moving  iqwn  a  plane  vertical  curve, 
let  V  denote  the  angle  which  the  radius  of  curvature  q  makes  with 
its  horizontal  projection,  and  the  equation  (SIT^)  gives 

^z=  — ^COSl^, 

which  is  the  equation  of  the  cycloid  referred  to  its  radius  of  curva- 
ture and  angle  of  direction,  so  that  the  cycloid  is  the  tautochrone  of  a 
free  heavy   body   in   a   vacuum.       The  same  curve,  drawn   upon  the   de- 


—  319  — 

veloped  surface,  is  the  taidochrone  of  a  heavy  lodfj,  moving  upon  a  vertical 
cylinder. 

541.  Every  curve  may  be  regarded  as  being  upon  the  surface 
of  its  vertical  cylinder  of  projection ;  and,  therefore,  the  taidochrone 
of  a  heavy  hody  moving  in  a  vacuum  upon  any  surface  luhatever,  is  the 
intersection  of  the  surface  with  such  a  vertical  cylinder,  that  the  intersection 
is  a  cycloid  upon  the  developed  vertical  cylinder.  The  determination  of 
the  tautochrone  upon  any  surface  is  thus  reduced  to  a  problem  of 
pure  geometry.  If  the  axis  of  z  is  the  upward  vertical,  and  if  z^  is 
the  height  of  the  lowest  point  of  the  curve  above  the  origin,  the 
equation  (SlTie)  becomes,  in  the  present  case, 

542.  If  a  heavy  hody  is  restricted  to  move  upon  a  cylinder  of  which 
the  axis  is  horizontal,  and  of  which  the  equation  of  the  base  is 

()j  =  na  cos  Vj  sin""-^  v-^ , 

in  which  v-^  is  the  angle,  which  the  radius  of  curvature,  denoted  by 
()i,  makes  with  the  upward  vertical ;  and  when  the  cylinder  is  devel- 
oped into  a  vertical  plane,  if  y  is  the  height  of  the  moving  body 
above  the  horizontal  line,  which  corresponds  to  the  lowest  side  of 
the  undeveloped  cylinder,  the  value  of  y  is 

y  ■=.a  sin "  i\ . 

The  force  of  gravity,  resolved  in  a  direction  tangential  to  the 
cylinder,  is 

^sinr'i==yy/^; 
so  that  the  present  problem,  corresponds  to  that  of  a  hody  moving  in  a  ver- 


—  320  — 

tical  plane,  and  subject  to  a  force  ivJiich  is  fixed  in  direction,  and  propor- 
tional to  some  poiver  of  the  height  above  a  given  level.  The  equation 
(319i3)  gives  for  the  equation  of  the  tautochrone 

543.  If  V  denotes  the  angle  which  the  radius  of  curvature  [{)) 
of  the  tautochrone  makes  with  the  upward  vertical  in  the  developed 
cylinder,  the  equation  (317i4)  gives 

2B 

sm  V  sin  v,  =  —  s. 
9 

which,  substituted  in  (32O5),  reduces  the  equation  of  the  tautochrone  to 

g        '  n-\-l      \gsva.  v/ 

544.  When  ^0  vanishes  in  the  problem  of  the  preceding  sec- 
tion, the  equation  of  the  tautochrone  becomes 


,    .    1+1  I      naq       /y\V±. 


n  +  l 


n+l  ,     .    -^ 

0  =  — ~  0  sni"-^  V 


or  ()  =  — —-  0  sni"-^  V  cos  v 

^         n  —  1 


in  which 


(^^r-'='^m 


so  that  the  tautochrone  on  the  developed  cylinder  of  §  542  is  of  the  same 
trigonometric  class  of  curves  uith  the  base  of  the  cylinder,  U'hen  it  passes 
through  the  lowest  side  of  the  undeveloped  cylinder.  This  case  is  impos- 
sible, when  n  is  included  between  positive  and  negative  unity ;  for 
when  n  is  negative  and,  independently  of  its  sign,  less  than  unity,  s 
becomes  infinite  when  y  vanishes,  but  when  n  is  positive  and  less 


—  321  — 

than  unity,  the  derivative  of  (32O19),  which  is 

D^S  =  COSeC  V  =  ^  y/  -jj  [-)  -An  , 

gives  the  impossible  result  that  cosec  v  vauislies  with  ?/. 

545.  The  differential  equation  of  the  tautochrone,  in  the  case  of  § 
542,  referred  to  rectangular  coordinates  upon  the  developed  cylinder,  is 
readily  obtained  from  the  equations  of  §  542,  which  give 

in  which 

7,2 0  "  +  ^ 

and  the  axis  of  x  is  horizontah 

In  the  case  of  §  544,  in  which  ^^  vanishes,  this  equation  becomes 


/?,.^+l=/r(f)--\ 


54G.  In  the  case  in  which  n  is  unit}',  that  is,  in  which  the  base 
of  the  cjjlinder  is  a  ctjcloid,  the  equation  of  the  tautochrone  on  the  developed 
cylinder,  becomes 

AYhen  z^  vanishes,  this  curve  is  reduced  to  a  straight  line,  but 
in  all  other  cases,  its  form,  if  it  is  infinitely  extended  in  the  plane 
of  the  developed  cylinder,  resembles  the  hyperbola.  By  the  adop- 
tion of  the  notation 

.   ^  .       2aB 

snr  I  =^ , 

it 

y=^{2az,)Bec(p, 

41 


322  

and  that  of  elliptic  functions,  its  equation  may  be  expressed  in  the 
forms 

^  =  y/^tan9), 

:r  =  J^-jj  (cos  &  tan  9)  +  ^z  9  —  ^i  9)  • 

547.  If  ci  heavy  hody  is  restricted  to  move  upon  a  surface  of  revo- 
lution about  a  vertical  axis,  of  which  the  equation  of  the  meridian 
curve  is  that  of  (319i-).  If  ^  is  the  distance  of  the  body  on  the 
meridian  curve  from  the  lowest  point  of  the  surface,  the  value  of 
y  is  given  by  the  equation  (olOgs),  and  the  force  of  gravity,  resolved 
in  a  direction  tangential  to  the  meridian  curve  is  expressed  by 
(SlOgg),  so  that  the  present  problem  resembles  that  of  a  body 
moving  in  a  plane,  and  subject  to  a  force,  which  is  directed 
towards  a  fixed  point  in  the  plane,  and  is  proportional  to  some 
power  of  the  distance  from  that  point.  The  equation  (olT^)  of  the 
taidochrone,  gives 

f,         ,,m  +  l ,,  jn  +  1 

p  ,.2 9       y         —  !h 

m  -j- 1  a'"  ' 

in  which  m  is  the  reciprocal  ofn,  and  y^  the  value  of  y  at  the  lowest 
point  of  the  tautochrone. 

548,  When  m  vanishes,  the  surface    of  revolution   is   a  rir/ 
cone,  and  the  equation  (322j9)  becomes 

Bs''==:y{y—y,). 

By  means  of  the  notation 

sec  p  =:  1  H -; 


the  angle  ((p)  which  ?/  makes  with  ?/q  in  the  developed  cone  is  given 
by  the  formula 

tan[(0  +  jy)tani|<(]  =  ^; 

SO  that  the  polar  equation  of  this  tautochrone  upon  the  developed  cone  is 
expressed  hy  the  comhinaiion  of  (32228)  and  (323o). 

549.  When  ^^  vanishes,  /i  also  vanishes,  and  the  equation 
(3233)  becomes 

^  +  ^y  +  cot^=o. 

550.  When  m  is  unity,  the  surface  of  revolution  is  cj'cloidal 
and  the  equation  (322j9),  becomes 

which  becomes  the  meridian  curve  itself,  when  y^  vanishes. 

•551.  In  the  case  given  in  (322^4),  of  a  body  moving  in  a  plane 
and  subject  to  a  force,  ivhich  is  directed  totvards  a  fixed  point  in  the  plane, 
and  is  proportional  to  some  potver  (m)  of  the  distance  from  that  jjoini, 
the  equation  of  the  tautochrone  may  he  given  in  the  form 

in   which   the  attracting  point   is   the  origin  of  polar  coordinates. 
The  polar  differential  equation  is 

552.  If  the  attraction  or  repulsion  of  the  point  had  hecn  any  function 
zvhatever  of  the  distance  from  the  origin,  the  equation  of  the  tautochrone 
tvoidd  have  assumed  the  form 


s^^Fr  —  Fr 


0? 


—  324  — 

in  Avliicli  F  denotes  the  function  of  which  the  derivative  expresses 
the  given  law  of  attraction.  This  equation  may  therefore  assume 
the  form 

in  which  S-^  is  a  function  of  S.  If  then  v  is  the  angle  which  the 
radius  of  curvature  makes  with  the  axis  of  x,  the  derivatives  of 
this  equation  are 

2  X  sin  V  —  2y  cos  v=z  S[, 

(2^ cos v-\-2i/  sin v) D,v  =  S['—  2  • 
whence 

2xD,v  =  S[  sin  v B.v -\-  {jS['  —  2)  cos  v, 

2^  B.v  =  —  >S[co^v  B.v  +  {jS['  —  2)smv, 

iS^Dy-=StDy-^{S"  —  2f, 

Avhich  is  the  equation  of  the  taidochrone  expressed  in  terms  of  the  radius 
of  curvature  and  the  arc. 

553.     The  polar  differential  equation  of  the  tauiochrone  in  the  case 
of  the  preceding  section  is 


r'D.if  +  l 


Fr—Fu' 


which  is  the  same  equation  with  that  wdiich  is  given  by  Puiseux. 
554.     The  derivative  of  (324ig)  relatively  to  v  is 

so  that  the  _  elimination  of  5  between  (324iq)  and  (32427)  gives  the 
differential  equation  of  this  tauiochrone  in  terms  of  the  radius  of  curvature 
and  the  angle  of  its  direction. 


—  o-::o  — 

555.  In  the  case  of  §  552,  when 

/S ]  =^  as  -\-  b, 
the  value  of  S^  is 

The  eqiiation  (32427)  becomes  therefore 

and  the  cqiiaUon  of  the  taidochrone  is 

^)  =  ^  av, 

21'hicJi  is  that  of  the  involute   of  the   circle.     This  case    corresponds  to 
that  in  which  the  law  of  the  central  force  is  of  the  form 

556.  In  the  case  of  §  552,  when 

>Si  =  «  (5  -f-  by, 
the  value  of  S^  is 

S,=  i/^m^{s  +  ^)\ 


in  which  n^ 


a 


I— a 


SO  that  cc  must  be  positive    and  less  than  unity.       The   equation   of 

the  taidochrone  is,  then,, 

which  is  that  of  the  logarithmic  spiral.     This  case  corresponds  to  that 
in  wdiicli  the  law  of  the  central  attraction  is  of  the  form 


—  326  — 

that  is,  in  ivlikli  the  force  is  proportional  to  the  distance  of  the  Ijody  from 
the  circumference  of  the  circle  described  from  the  origin  as  the  centre  ivith 
a  radius  erpial  to  that  of  the  initial  position  of  the  hody.  Tliis  case  is 
discussed  by  Puiseux. 

557.  In  the  case  of  §  552,  ivhen  the  force  is  proportional  to 
the  distance  from  the  origin.  The  equation  (3233i)  assumes  the 
form 

.2  _  r'—rl 
a      ' 

which,  with  the  value  of  m  in  (32522),  reduces  S^  and  S^  to 

The  equation  of  the  tautochrone  is,  therefore, 

of  which  the  integral  is 

Q  =  ,    "     Cos  (m  v) 

^         1  —  a  ^        ' 

in  which  the  arbitrary  constant  is  determined  so  that  v  may  vanish 
with  s. 

The  second  derivative  of  this  equation  gives,  for  the  radius  of 
curvature  of  the  second  evolute  of  the  tautochrone 

so  that  the  second  evolute  is  similar  to  the  tautochrone  itself 

In   the   case   in   ivhich  m  is  real,  which  corresj^onds  to  that   in 


327 


which  a  is  positive  and  less  than  unity,  this  curve  runs  off  to 
infinity  in  each  direction,  with  a  constantly  increasing  radius  of 
curvature. 

In  the  case  in  ivhich  m  is  imaginavfj,  the  substitution  of 

2  2 

— •  n  =  ntr , 
reduces  the  equation  of  the  tautochrone  to  the  form 

()=  —-^cos  (nv). 

zvJiich  is  the  equation  of  an  epicycloid.  The  epicycloid  is  formed  hy  the 
external  rotation  of  one  circle  upon  another,  ivhen  n  is  less  than  unity,  in 
ivhich  case  a  is  negative  and  the  force  is  repulsive ;  hut  the  epicycloid  is 
formed  hy  internal  rotation,  when  n  is  greater  than  unity,  u'hich  corresponds 
to  the  case  when  a  is  positive  and  greater  than  unity.  In  either  of  these 
cases,  the  initial  velocity  must  not  be  more  than  sufficient  to  carry 
the  body  to  either  of  the  cusps. 

In  the  case  in  ivhich  a  is  infinite,  the  tautochrone  is  reduced  to  a 
straight  line. 

The  example  of  this  section  is  discussed  by  Puiseux. 

558.  The  examj^le  of  the  preceding  section  embraces  the  case 
of  any  force,  which  is  a  function  of  a  distance  from  the  origin,  in  the 
immediate  vicinity  of  the  point  of  greatest  velocity.  The  form  of 
the  tautochrone,  near  the  point  of  greatest  velocity,  in  the  example  of 
§  552,  is  typified,  therefore,  hy  the  epicycloid,  or  hy  the  curve  of  erpia- 
tion  (32621). 

559.  The  hivestig-ation  of  the  tautochrone  in  a  resisting; 
medium  is  postponed  to  the  general  case  of  the  chronic  curves. 


—  328 


THE     BRACHTSTOCHRONE. 

5G0.  The  curve  upon  wliicli  a  body  moves  in  the  least  pos- 
sible time  from  one  given  point  to  another,  is  called  the  hrachjs- 
tochrone. 

561.  The  investigation  of  the  general  case  of  a  brachysto- 
chrone  which  is  confined  to  any  surface  or  limited  by  any  condition, 
may  be  conducted  by  means  of  rectangular  coordinates.  The  time 
of  transit  from  the  first  to  the  last  of  the  given  points  may  be  ex- 
pressed by  the  equation 

which  is  to  be  a  minimum.     This  condition  gives,  for  each  of  the 
other  axes,  the  equation 


A(^)-Ai>,.(^)  =  o. 


562.  When  the  body  is  only  subject  to  the  action  of  fixed 
forces,  V  does  not  involve  either  t/'  or  /,  and  the  preceding  equation 
becomes 


D,. 


or  by  (316i;), 

i),i2  +  t.^A(^')=0. 

563.     If  the  plane  of  x  i/  is  assumed,  at   each  instant,  to  be 
that  in  which  the  body  moves,  and  if  the  axis  of  u  is  taken  normal 


—  82y  — 

to  the  path  of  the  body,  the  preceding  equation  becomes,  if  o  ex- 
presses the  radius  of  curvature  of  the  path 

so  that  ilie  ccninfvgal  force  of  iJie  hodij  h  equal  to  the  normal  jivcssnre, 
and  the  whole  2yi^^ssi[re  upon  the  hrachf/stochroue  is  double  the  centrifugal 
force.     This  proposition  was  discovered  by  Euler. 

06 4.      When  the  normal  pressure  vanishes,  the  radius  of  curvature 
is  infinite,  which  corresponds  in  general  to  a  point  of  contrar//  flexure. 
-  When  there   is   no  force  acting  upon    the  hodg  throughout   its  patli,  the 
hrachgstochrone  is  reduced  to  a  straight  line. 

565.  Any  conditions  to  which  the  path  must  be  subject, 
whether  elementary  such  as  that  it  is  confined  to  a  given  sur- 
face, or  integral  such  as  that  its  whole  length  is  given,  must  be 
combined  with  the  general  condition  of  brachystochrouity  hy  the 
usual  methods  of  the  calculus  of  variations, 

566.  If  the  only  force  u'hich  acts  upon  the  hodg  is  directed  to  a  given 
point,  and  if  the  path  is  sul)ject  to  no  conditions,  \Qt  the  plane  of  .z-.e-  be 
assumed  to  be  that  which  passes  through  the  centre  of  action  and 
the  initial  element  of  the  path.  In  this  case  the  equation  (02827) 
gives 

cos^  =  0,     ^=^:t, 

or  the  hrachgstochrone  is  contained  in  a  plane  v:hich  passes  through  the 
centre  of  action. 

567.  The  preceding  case  includes  that  in  which  the  centre 
of  action  is  removed  to  an  infinite  distance,  so  that,  in  the  case  of 
'parallel  forces,  the  free  hrachgstochrone  is  contained  in  a  plane,  which  is 
parallel  to  the  direction  of  the  forces. 

568.  When  the  hodg  is  acted  upon  hy  no  forces,  or  only  hy  those 
ivhich   are    normcd  to  its  path  and  do  not  tend  to  change  its  velocity,  the 

42 


—  330  — 

equation  (32813)  shoivs  that  the  hr achy dochr one  is  the  shortest  line  which 
can  lie  drawn  under  the  given  conditions. 

569.  When  the  force  is  directed  towards  a  fixed  centre,  the 
equation  (320,,),  combined  with  (31Gi8)  gives,  if  the  centre  is  adopt- 
ed as  the  origin 

D,^    2 

If  p  is  the  perpendicular  let  fall  from  the  origin  upon  the 
tangent  to  the  curve,  this  equation  becomes 


Si  —  Sio         ^"^^i" 
of  which  the  intcixral  is 


wliich  is  the  equafion  of  the  hrachystochrone  referred  to  the  radius  vector 
and  the  loerpendicular  front  the  origin  ujwn  the  tangent  as  the  coordinates. 
This  form  is  given  by  Euler. 

570.      When  the  force  in  the  preceding  case,  is  proportional  to  the 
distance  fro)n  the  origin  so  that  11  has  the  form 

n^ar'', 
the  equation  (33O14)  becomes 

of  which  the  derivative  jj;ives 

P 

If  V  is  the   angle   which  {)  makes  witli  the  fixed  axis,  the  de- 


ooi    

rivative    of   this    last    equation    gives,    by    means  of  the  preceding 
equation 

''A  ^v  V  =  ''  cos  :=\  [_'(}r,{\  —  af,)  o-  +  rfj  , 

which  becomes 

if 

2         1  —  a  pi 
a  PI 

The  integral  of  this  equation  is 

0  =:  — —,  Sin  (ni  v) 

^         m  u  j)i  ^         ' 

SO  that  its  second  cvoluic  is  similar  to  the  hrach/jstochronc  itself. 

When  m  is  real,  which  corresponds  to  the  case  of  a  repulsive 
force,  and  ap\  less  than  unity,  this  hrach>/stoehrone  is  a  spiral  which  has 
a  ciisjj  at  the  point  at  n'hivh  v  vanishes. 

When  m  is  imaginary,  the  substitution  of  (32T5)  reduces  (331n) 
to  the  real  form 


■7.  sin  {n  v) 


napi 

so  that  in  this  case,  the  hrachystochrone  is  an  epicycloid  jrhich  is  formed  hj 
internal  rotation  when  the  force  is  attractive,  and  hjj  external  rotation  when 
the  force  is  repulsive.     This  case  is  given  by  Euler. 

571.      ^yhcn  the  forces  are  p>arcdlel,  ihQ  equation  (0293)  gives,  if 
the  axis  of  *  is  supposed  to  be  in  the  direction  of  the  forces 

^'^  =4,=2cot=A:, 


^ ^^,  QSm 


of  which  the  integral  is 


r> n 

-^0 


in  which  a  is  an  arbitrary  constant,  and  this  is  the  equation  of  the 
hracht/stochrone  referred  to  the  coordincdes,  ivhich  are  z  and  the  inclina- 
tion of  the  curve  to  the  axis  of  z ;  and  the  equation,  referred  to  q 
and  I  as  coordinates,  is  ohtaincd  hy  eliminating  z  hetween  (03I27)  and 
(33I31). 

572.     In   the   case  of  a  constant  force,   the    preceding   equation 
assumes  the  forms 


(J  (z  —  z^)  =  a  sin 


2«   .     ^ 

so  that,  in  this  case,  the  hrachystochrone  is  a  cycloid. 

573.  ^Yhen  the  imrallel  forces  arc  iwoportional  to  the  distance  from 
a  given  line,  which  may  be  adopted  for  the  axis  of  x,  the  vaUie  of 
il  has  the  form 

wdience  the  equation  of  the  hrachystochrone  is 

a  sin  s 

When  the  force  is  repulsive,  or  luhen  it  is  attractive,  hut 

this  curve  consists  of  hranches,  ivhich   are  united  hy  cusps,  and  resemhle 
the  cycloid  in  general  form  ;  hut  tvhen  the  force  is  attractive,  and 

this  curve  consists  of  hranches   which  are  still  united  hy   external  cusps ; 
hut  the  middle  point  of  each  hranch  is  upon  the  axis  of  x,  and  is  a  point 


of  inflexion,)  and  the  interval  hctiveen  tivo  successive  points  of  inflexion,  cx- 
jjressed  hij  elliptic  integrals,  is 

v/(-i)[^.(i^0-9^.(J^)], 
in  Avliich 

sin  1:=.  z,^\J , 

^  a 

In  the  case  of  the  attractive  force,  and 

I        a 

the  equation  of  the  Irachgstochronc  becomes 

()  — .ejan^, 

tvhich  consists  of  two  inflnite  branches  joined  hj  an  external  cusp,  and  the 
axis  of  X  is  an  asymptote  to  each  of  the  branches. 

574.  When  the  body  is  subjected  to  move  upon  a  given  sur- 
face, the  force  by  which  it  is  retained  upon  the  surface  is  joerpen- 
dicular  to  its  path,  and  must  be  united  with  the  second  member  of 
equation  (8293).  Hence  it  follows  that  the  centrifugal  force  of  the 
body  in  the  direction  of  the  tangent  plane  to  the  surface,  upon  ivhich  it  is 
conflned,  is  equal  to  the  normal  force  ivhich  acts  in  this  plane  normal  to 
the  brachjstochrone. 

At  the  beginning  of  the  motion  when  the  velocity  is  zero, 
there  is  no  centrifugal  force,  so  that  the  initial  direction  of  the 
br achy stochr one  upon  the  surface  coincides  vAih  that  of  the  tangential 
force. 

575.  If  the  first  and  last  points  of  the  brachystochrone  are 
so  situated  upon  the  given  surface,  that  a  line  can  be  drawn 
through  them,  which  coincides  throughout  with  the  direction 
of  the  tangential  force  to  the  surface,  this  line  is  the  brachysto- 
chrone. 


OO-i     

Hence,  the  hrach/jstochrone  upon  the  surface  of  revolution  is  the 
meridian  line,  ivhen  both  its  extremities  are  upon  the  same  meridian  line, 
and  the  force  is  directed  to  a  point  upon  the  axis  of  revolution,  or  is  parallel 
to  this  axis. 

576.  In  the  general  case  of  a  surface  of  revolution  and 
a  force  which  is  directed  to  a  point  upon  the  axis  of  revo- 
lution, let 

o  denote  the  arc  of  the  meridian  curve  measured  from  the  pole, 
u   the  perpendicular  from  the  surface  upon  the  axis, 
^)^  the  radius  of  curvature  of  the  projection  of  the  brachjstochrone 
upon  the  tangent  plane  to  the  surface, 

and  the  proposition  (SoSgo)  is  expressed  by  the  equation 

-=Z>.i2tan'! 


which  gives 

But  the  equations 


X>,i2  D,{i-)       2  cot? 


il il^i  V'  Qt 


give 


Cr      ' 


D^u  =  cos "  =  cos  1  cos  Z, 

j^  ^ 1         sin  f  cos  ^ 

i>T  U  ^ 

D,{ud\\l)- 

and  if  ^1  is  an  arbitrary  constant, 

Ds  log  V  =  ^<,log  [u  sin  %) 
A  z'  =  u  sin^==«^  A" 
A  v^  =^  u  V  sin  1  =  ic"  Dt  I 

so   that   the   area   described  by   the  projection  of  the  radius  vector  upon 


^  335  — 

ihe   'plane    of  x  y  is  projiGrtional  to    the  square    of  the   vclocify   of  the 
hodij. 

577.     The  equation  (33498)  gives 

JJ„s^=^  sec  1 


„^tan?^  Av  _A    I        2(0— .Q,,) 

^  ■'•  ?«  M  v'  (it-  —  A'v')        u  V  ?t-  —  2  ^^  {SI  —  Sio) ' 

578.     If  (5  is  the  angle  which  the  radius  rector  makes  with  the 
axisj  the  preceding  values  give 

n  ,  —  J  «^['-!+j(^^ '■)•-'] 
-^^  ^  ~  V  «'-— 2  A^  (Si—  si,y 

9^       «-y      M-  — 2.1- (i2  —  iio) 
When  the  forces  are  parallel  these  equations  give 


A  D,G    I        2  (.Q  —  /2„) 


w       V    « 


(^  _  2  A  {.Q  —  iio) ' 


579.  .  Upon  the  surface  of  revolution  which  is  determined  hy 
the  equation 

B  v=t(, 

in  which  B  is  an  arbitrary  constant,  the  value  of  1  is  by  (3342^) 
constant,  so  that  upon  this  surface  tlie  hraciiystochrone  niaJces  a  constant 
angle  with  tJie  meridian  curve.     In  the  case  in  which 

A=iB 

the  brachystochrone  becomes  perpendicular  to  the  meridian,  and  is 
a  small  circle,  of  which  the  plane  is  horizontal. 

Whatever  is  the  value  of  B,  the  point    at  which  v  vanishes, 


—  336  — 

coincides  with  that  at  which  ii  vanishes,  so  that  at  the  pole  of  this 
surface  the  velocity  vanishes. 

Upon  any  oilier  surface  of  revolution  about  the  same  axis,  the  incli- 
nation of  the  hr achy stochr one  to  the  meridian  arc  is  the  same  ivith  the 
corresponding  inclination  upon  the  surface  of  equation  (33592),  ^'^  ^^^^  com- 
mon circle  of  intersection  of  these  two  surfaces.  Hence  the  limit  of  the 
hrachystochrone  upon  a  given  surface  of  revolution  is  its  circle  of  intersection 
ivith  the  surface  of  equation 

Av  =  u , 

and  the  hrachystochrone  extends  over  that  portion  of  the  given  surface, 
ivhich  is  exterior  to  the  given  surface,  by  which  the  limits  are  thus  defined. 

580.  In  the  case  of  a  heavy  body,  the  surface  of  equation 
(33592)  is  a  paraboloid  of  revolution.  When  the  velocity  of  a  heavy  body 
upon  any  paraboloid  of  revolution,  of  tvhich  the  axis  is  vertical  and  directed 
dowmvards,  is  just  sufficient  to  carry  it  to  the  vertex,  the  hrachystochrone 
maJces  a  constant  angle  ivith  the  meridian  curve;  but  ivhen  the  velocity  is 
too  small  to  carry  the  body  to  the  vertex,  the  hrachystochrone  is  a  curve 
ivhich  maJces  an  increasing  angle  ivith  the  meridian  as  it  descends,  and  may 
sometimes  become  perpendicular  to  the  meridian ;  and  luhen  the  velocity  is 
more  than  sufficient  to  carry  the  body  to  the  vertex  of  the  paraboloid,  the 
hrachystochrone  is  an  infinite  curve,  ivhich  is  horizontal  cd  its  highest  point, 
and  diminishes  its  angle  ivith  the  meridian  as  it  descends. 

If  the  equation  of  the  paraboloid  is 

l\^  z=:z  4:  p  ;S 

in. which  the  axis  of  3  is  the  downward  vertical,  the  equation  (33408) 
becomes 

If  .^0  is  positive  and 

p>iA^g, 


—  oo7  — 

the  substitution  of 

•   2  ^'ff 

sin-"  a  =  — -^ , 
2p 

q  ^=  Sq  tan^c?, 

^  —        p  —  q 

gives 

s  z=  J  sec  a  {p  —  rj)  ((p  +  Sin  (p), 

in  which  the  upper  signs  correspond  to  the  case  in  which  ])  is 
greater  than  </,  and  the  lower  to  that  in  which  p  is  less  than  q. 
In  the  case  in  which  p  is  greater  than  q,  the  substitution  of 


cos'  1/' 


■+P' 


snr^  =' — —^, 
gives 

^  =  —  tan  « 4  /  n  -|-  -  j    7fi  1/ '  —  ^ ,  If  —  cot  «/^  y/  ( 1  —  sin^ «  sin^  i/^ ) 
When  2^  i«  smaller  than  q,  the  substitution  of 

2  ~  ~0 

COS""  L"  ^  — , , 

gives 

'^  =  tan  a  t  /  (^^^')  I  ^i  f  — -  '3=^  i/'  +  cot  i/'  v'  (1  —  sin-/  sin^  i/') 

When 

43 


oo; 


the  arc  is 

s  =z  sec  a  {z  —  Zq)  , 

so  that  its  indincdion  to  the  axis  is  constantly  equal  to  a,  and  the  brachj'S- 
tochrone  is  defined  by  tlie  equations 

z  =  2'o  sec^  cj) , 

'^=tana^/|'(tang)  — 9). 
When 

the  arc,  measured  from  its  cusp,  is 

2 

and  if 


[i^+p?-{^o+py^] 


tan  J' ==*/-, 

the  brachj'stochrone  is  defined  by  the  equations 
p  -\-  z=:p  sec^ ;'  Cos^  (p , 

tan  y  \2  sui  Z  y    '    tan  2  y         / 

when 

p<iA'y, 

in  which  case  the  brachj'stochrone  has  a  lower  limit  at  which  it  is 
horizontal,  the  substitution  of 

sec  a  =  — — , 

2p 
q  = 


sin-  a' 


2z-\-p  —  q 

cos  W  = -J- . 


ooJ    

gives,  at  the  lowest  point  of  the  curve,  Mhere  (p  vanishes 

and  for  the  value  of  s,  measured  from  the  lowest  point, 

^  =  2  (;>  +  :?)  cot  a  (sin  (f  +  f|). 
The  substitution  of 

tan-  w  = , 


•    9  •         <7 

sni- 1  — 


P  +  9 
tjives 


iin  a  sj[p{p^rj)} 


When  2-0  is  negative,  in  which  case  the  condition  (00630)  is 
satisfied,  the  substitution  of  the  equations  (00T2--5)  with  the  lower 
sign  gives  the  corresponding  value  of  (ooTj)  for  the  arc  measured 
from  its  upper  limit,  which  corresponds  to  the  vanishing  of  9). 

When 

the  substitution  of 

cos-  w  =  — f^ , 

.   9  ,       7'  +  ~o 

snr  I  ^  — — 

jj  —  (J 

gives 

;;  =  tan  a  y/  (^^)  \^i  'H'  —  3=^  1//  +  cos  ij'  \  (cot-  if  -j-  cos"  /) 

i^g>,( ^,„.)]. 

-r/         \       p  —  (/     ■  /J 


'    p—q  \       p—q 

When 


—  •^o>7^ 


the  sub.stitution  of 


—  340  — 


2  -  +  <? 

COS''  Ij!  =  ^— ^ 


sin  I  — 


gives 

^  =  tan  a  J  y     ^"  ~  ^  j    H^,  t//  —  S'^.  V'  +  ^os  i/^  y^  (cos^  iji  -f-  cos^  ^) 

When 

the  braehystochrone  is  defined  by  the  equation 

:  =  tan  a  \J C-±A  +  H /^  log  ^ rPl^A ' 

581.  7;i  t/ie  case  of  ike  heavy  hody  upon  the  j^o^raholoid  of  revolu- 
tion in  2vhich  the  axis  is  vertical  and  directed  upivards,  the  braehystochrone 
forms  an  increasing  angle  with  the  meridian  as  it  descends  and  is  perpeii- 
dicular  to  the  meridian  at  its  lowest  point.  In  this  case,  the  incUnation 
to  the  meridian  is  determined  by  the  equation 

if  (33625)  is  the  equation  of  the  paraboloid.     By  the  substitution  of 

•  2  ^'H 

^wY  a  =  — ^ , 

2p 
??  =  ^0  tan^  a , 

Cos  (T)  =^  -i , 

(p  vanishes  at  the  lowest  point  where 


—  341  — 

and  the  value  of  the  arc,  measured  from  the  lowest  point,  is 

s  =  i  (p  -\-  rj)  sec  a  [(f>  -\-  Sm  (/  ) . 
The  substitution  of 

tan-"  L"  = , 


•  2  •       ^0  —  q 
snr^^     -, — -, 


srives 


582.     In  the  case  of  a  licavij  hod//  upon  a  vertical  rujht  cone,  if  the 
vertex  of  the  cone  is  assumed  as  the  origin,  and  if 

a  is  the  angle  which  the  side  of  the  cone  makes  with  the  axis, 

A'  q  CO?  a 

^  =^  the  angle  which  r  makes  with  the  axis  upon  the  developed 
cone, 

the  inclination  to  the  meridian,  the  derivative  of  the  arc  and  of  ^  are 

Sin  1  =  ^ '-^^ —^  , 

D.s 


jy^^        v/[2.,(;--n,)] 

r  y/  [r- —  2  r^  (r  —  r^)]  * 

When 

2ro>ri, 

the  substitution  of 


snr2 

'J. ) 


r,  cot  I 


tan  w  =: 

^  r  —  r  J 


r  =  i-Qsec^  Ig), 


—  342  — 

gives 

cos  ^=  cos  ^  sec  (i/'  —  /), 

s  =  Tq  sin^ '/  (cosec  iji  —  cosec  2  z)  —  r^  log  (tan  ^  iji  cot  ?") , 
^  =  —  siii^"-^^  (sin  /  sin  (p)  -\-  sin  i '^\  (/> , 

in  which  the  arc  is  measured  from  the  cusps,  at  which  point 

TJiis  Inichjdochronc  extemls  to  infinitfj  from  the  cusjj  tvitJiont  ever  Itecoming 
peiyendicuhw  to  the  side  of  the  cone.  The  greatest  angle  which  it  makes 
with  the  side  is  /,  and  at  this  point  of  least  inclination  to  the  side 

yi  =  i,     r=2ro,     (p=^^7i, 
6=-  —  i-\-  sin '/  ^  j  ( ^  71 ) . 
When 

the  hrachjstochrone  is  defined  hij  the  equation 

J  <5  =  ta„-y  C;^  - 1)  -  Cot-y  (1  - 1) , 

and  the  length  of  the  arc,  measured  from  the  point  of  least  inclina- 
tion to  the  side,  is 

5  =:  r  —  2  To  +  r,  log  {{^  —  l)^ 
When  ;■()  is  positive  and 

the  substitution  of 

9  r 

Sec^P*  =-^^-^, 


^  I   r,  Tan  3 

1  an  ii;  r=  + '-  , 


—    o4o   

gives 

s  =  j\  Tan  [}  Cosec  !/'  —  rx  log  Tan  ^-  i/^ , 

in  which  the  arc  is  measured  upon  each  branch  from  the  point  at 
which  it  is  horizontal  and  the  upper  sign  belongs  to  the  lower 
branch  and  the  reverse.  T/ie  vpjjer  branch  is  finite,  ivhilc  the  lower 
branch  is  infinite,  and  the  value  of  i/'  extends  on  the  upper  branch 
from  2  [)  to  infinity,  and  on  the  lower  branch  from  inhnity  to  zero. 
For  the  upper  branch  the  substitution  of 

sin  i:^e~'^'\ 
r  —  ro  =  To  sin  i  sin^  f/) , 
gives 

^  =  2(l  +  sinz)[3^,9-^,(sin/,(p)]. 

Upon  the  lower  branch  the  substitution  of 


r  —  ^o  =  -. — .  .  .,    , 

sill  I  i<UV-  Vj 

gives 

(3  =  2(14-sin«")'5\(sin/,i//). 

When  Yq  vanishes,  the  ecjimtion  of  the  brachjstochrone   iipon   the   de- 
veloped cone  is 

r  =  2ri  sec^  h.  ^, 

and  the  leni>-th  of  the  arc  is  • 

5-  =  2  y-i  tan  -^  t5  sec  i  (^  +  2  r^  log  tan  ( i  tt  +  i  (^ ) . 
When  ^0  is  negative,  the  substitution  of 

^  9  ,j  2  ?n  —  4  sin  % 

Cosec"  5  =  —  —  = 


i\  (i  -j-  ^i'l  0^ 

^  —  r,  Cos/i' 


,  V()^  ^1 


—  344  — 

gives 

6'  =  i'l  Cot  (■)  Cosec  If  —  Vi  log  Tan  |  i/' , 

in  which  the  order  of  the  signs  and  of  the  value  of  if  is  the  same  as 
in  (3484)  with  reference  to  the  branches.  27ie  upper  and  finite  branch 
of  the  t)r achy dochr one  lies  in  this  case  upon  the  upper  and  inverted  portion 
of  the  cone.  The  formula?  (343ii,  343i6_i9),  J^pply  to  this  case,  in 
which  it  must,  however,  be  noticed  that  the  sin  i  is  negative. 

583.      When  the  solid  of  revolution  upon  ivhich  the  heavy  hody  moves, 
is  the  ellipsoid  of  which  the  equation  is 


(z)+a)=i> 


the  inclination  to  the  meridian  is  determined  by  the  equation 

The  problem  naturally  divides  itself  into  two  cases.  In  the  first  case 
the  velocity  is  more  than  sufficient  to  carry  the  hody  to  the  highest  point  of 
tlie  ellipsoid,  tlie  hracJiystocJirone  is  a  continuous  curve  tvJdcJi  is  liorizontal  at 
its  liKjlicst  and  lowest  limits,  and  tvJiich,  always  running  round  ttie  ellipsoid, 
is  most  inclined  to  tlie  meridian  curve   at  tlie  point 

In  the  second  case,  the  velocity  is  not  sufficient  to  carry  the  hody  np  to 
the  highest  point  of  the  ellipsoid,  and  the  hr achy stochr one  is  horizontal  at 
its  lowest  point,  hut  has  cusps  for  its  nj)per  points.  In  each  of  these 
cases  the  length  of  the  arc  can  be  found  by  means  of  elliptic 
functions.  If  in  the  first  case  — .<i  and  >'.2  are  the  coordinates  of  the 
upper  and  lower  limits,  or  of  the  common  intersections  of  the 
ellipsoid  wdth  the  paraboloid  of  revolution  of  which  the  equation  is 

n'  =  2A'g(z-z,), 


—  345  — 

and  if  in  the  second  case  z^  refers  to  the  intersection  of  the  ellipsoid 
with  the  paraboloid,  while  —  z^  is  the  coordinate  of  the  intersection 
of  this  paraboloid,  inverted  at  the  horizontal  plane  of  ux^  with 
the  hyperboloid  of  revolution,  of  which  the  equation  is 

(i)-(zj=i. 

the  derivative  of  the  arc  is 

In  the  first  case,  ivhen  the  ellijosoid  is  p'olate,  and 
the  substitution  of 


gives 


2^ -r     - 

Cosy—    ~     ^\   "', 


s  =  -f  (^2  —  ^i)  ( Sin  g)  +  9) ) . 


When  the  ellipsoid  is  a  sjphere,  of  which  the  radius  is  R,  the 
hyperbola  (SlSg)  becomes  equilateral,  and  the  length  of  the  arc, 
measured  from  the  lowest  point,  is  determined  by  the  equation 


COS      — 


R—       z,-^z,       • 

In  the  first  case  (344i7),  the  substitution  of 


sintf;i 
COS«  =  -^ — -, 

44 


—  346  — 


gives,  for  the  sphere, 

„ 2  cos"-^  2  ^'i  Tii   /^^^  V^2  -\-  30S  \pi     s 


sin  n>- 


^'i  Tji   (^^^  V^2  +  30S  li^j      5   \    1^  2  sin^  ^  Ti),  (jp  /       cos  Ti^2  +  cos  ipi     s   \ 
~      i  \     1  _  cos  \v7~  '  2^/  ""        sin  1/;.  *  \  1  _j-  cos  V;^     '  2^/ 


4(,'0S"^  U'l  :;)7)  A'OSX/^-f-COSli'i        5    \  COS  1i>i COS  Ip^  CTj;    /    *    \ 

sin  u^o        *  \     1  —  cos  XI;.,     '  2  ^/  sin  if  o  *  \2  E/ 


tan^-i] cosii;2  +  cosT/;j 


\J  (sin"  1/'^  cosec-  ^  -\-  sin-  iX/i  sec^  gl^) 

In  the  second  case  (34423),  the  substitution  of 

2             2  z  —  Zo  —  Zq 
(p  =z ", 

sm  I 


^2  +  2^1 

gives,  for  the  sphere, 

„ Zj-\- R  ,^^  /cosxl'o  —  COSl/»o        \  Zi E  (^  /cosxl'o COSti'i         \ 

^       E  sin  U)^     *  V     1  —  cos  t!)2     ^  ^        Esinxp2        \     1  +  cos  w.^      '  '  / 

^1+^  \yn  (coi  'P2  —  cos  x/>o      \ gj    /i?  (1  —  cosi/»,)       \1 

i2cosii;,L     'V     1— cosi/»2     '^/  ^  \2,  +  i?cost//2  '  "/J 

I    cos  i/'o  —  cos  1'/.,  rrf=  ..       r    11  sin  { sin  q? 

-\ ^A ~  9=:  fp  —  cosec  I  tam~^J  -7— -i ^     .,    v . 

'  sin  1//2  ^  V  (1 4- COS-*  tan- q[.) 

In  tlie  case  in  which 

^0=  —  ^, 

the  brachystochrone  is  defined  by  the  equation 


s 


^^  2^    '  tan^i/>2 

584.     In  the  case  of  a  Iieavy  body  upon  any  surface  whatever, 
it  follows  from  (3203)  that 

V-         2r/(z  —  z,) 


6^i 


If,  then,  i\^^  is  the  normal  to  the  bmchystochrone  drawn  in  the 
tangent  phane,  and  extended  to  meet  the  horizontal  pLane  from 
which  the  body  must  fall  to  acquire  its  velocity,  the  preceding 
equation  gives 

i\;=(^  — 2-o)sec;^=^(^^, 

or  ihe  tangential  radius  of  curvature  of  the  hrachf/dochrone  is  twice  the 
icmgential  normal  ivhicJi  extends  to  the  horizontal  plane  of  evanescent 
velocity.     This  proposition  is  given  by  Jellett. 

585.  When  the  force  is  parallel  to  the  axis  and  proportional  to  ihe 
distance  from  a  plane  ivhich  is  perpendicular  to  the  axis,  the  surface  of 
revolution  of  erpiaUon  (33022)  i-^  (^ti  ellipsoid  v:hen  the  force  is  attractive 
towards  the  plane,  and  it  is  an  hgperholoid  of  two  sheets  when  the  force 
is  repulsive  from  the  plane. 

586.  ^Vhen  the  force  is  directed  tovmrds  a  fixed  point  and  propor- 
tional to  the  distance  from  the  point,  the  surface  of  equation  (33522)  ^'^  ^'* 
ellipsoid  if  the  force  is  attractive,  hut  if  the  force  is  repulsive,  the  surface 
may  he  an  ellipsoid  or  it  may  he  an  hyperholoid  of  tvjo  sheets. 

587.  When  the  force  is  directed  towards  a  fixed  point,  and 
inversely  proportional  to  the  square  of  the  distance  from  the  point, 
the  surface  of  revolution  of  equation  (33522)  is  defined  hy  an  equa- 
tion of  the  form 


=  aQ—1). 


588.  Other  conditions  might  be  combined  with  that  of  the 
brachystochrone.  Thus  if  the  total  length  of  the  arc  is  given,  the  normal 
pfessui^e  to  the  hrachystochrone  is 

D^n  =  '^^^^=-{i+hv), 

in  which  h  is  an  arbitrary  constant,  and  is  dependent,  for  its  value. 


—  348  — 

upon  the  given  length  of  the  arc.    This  constant  is  generally  infinite, 
when  the  brachystochrone  is  a  straight  line. 

589.  Under  the  condition  of  the  preceding  section,  the  equation 
of  the  hrachf/siochrone,  in  the  case  of  §  569,  referred  to  the  coordinates  of 
(33O17)  is 

In  the  case  of  §  570,  this  equation  gives 

590.  In  the  case  of  the  parallel  forces  of  §  571,  (34728)  P^(^s 

"  M  —  oasmy 

When  the  force  is  constant,  this  equation  gives 


so  that  when 

the  curve  has  points  of  contrary  flexure. 

591.     In   the   case   of  §  576,  and  ivith  the  condition  of  §  589,  the 
equation  of  the  hrachystochrone  has  the  form 


l-\-bv 

The  inclination  of  the  curve  to  the  meridian  arc  is  therefore  con- 
stant upon  the  surface  of  revolution,  which  is  defined  by  the  equation 

Bv=zu{l+l?v), 

and  this  surface  has  the  same  relation  to  other  surfaces  of  revolution  in 


—  349  — 

respect   to  tJie  hrachjstochrone  formed  under   the  present  conditions  vrith 
those  ivhich  are  indiccded  for  the  surface  of  \  579. 

In   the   case   of  a  heavy  body,  the  equation  of  this  defining 
surface  of  revolution  is 

,2 


2g(z-z,)  =  {-j^^ 


592.  If  the  condition  is  a  mechanical  one,  such  that  the  total 
expenditure  of  action,  defined  as  in  §  308,  shall  he  given,  the  normal  pres- 
sure to  the  hrachjstochrone  is 


DM  =  - 


1-2     1+5^2 


l  —  bv-^ 


in  which  h  is  an  arbitrary  constant,  and  is  dependent,  for  its  value, 
upon  the  given  expenditure  of  action.  When  this  constant  is  in- 
finite, the  normal  pressure  is  equal  and  opposed  to  the  centrifugal 
force. 

It  is  apparent,  from  the  preceding  equation,  that  under  the 
action  of  finite  forces,  this  brachystochrone  cannot  be  a  continuous 
curve,  in  one  portion  of  which  the  direction  of  the  normal  pressure 
coincides  with  that  of  the  centrifugal  force,  and  is  opposed  to  it  in 
another  portion. 

593.  Under  the  condition  of  the  preceding  section,  the  equation 
of  the  hrachystochrone,  in  the  case  o/"  §  569,  referred  to  the  coordinates  of 
(33O17)  is 

/2  — i2o    _  —(Py^ 

[\J^2b{^  —  fi,)j       \pj' 

In  the  case  of  §  570,  this  equation  gives 

^  1  —  2ba(r^ — r^)    '     Pisfoi 

594.  In  the  case  of  the  parallel  forces  of  §  571,  (3492,i)  gives 

—  sm^  ^, . 


[I -{-2b  (Si  — Si,)  J 


—  350  — 
When  the  force  is  constant,  tliis  equation  gives 

595.     Ill   the  case   o/"  §  576    and  tvith   the  condition  of  §  592,  the 
'equation  of  the  J)vach>jdochrone  has  the  form 


The  inclination  of  the  curve  to  the  meridian  arc  is,  therefore, 
constant  upon  the  surface  of  revolution,  which  is  defined  by  the 
equation 

and  this  surface  involves^  for  the  present  case,  the  iiroperties  of  the  defining 
surface  of  §  579. 

In  the  case  of  a  heavy  body,  the  equation  of  this  defining 
surface  of  revolution  is 

2  i?V  (^  -  ^«)  =  «^  [1  +  2  %  (2  -  z,)-\\ 

596.  The  hr achy stoclir one  in  a  medium  of  constant  resistance  is 
entitled  to  special  consideration.  In  this  case,  it  is  convenient  to 
introduce  the  length  of  the  arc  as  the  independent  variable.  The 
equation  of  motion  along  the  curve  is 

v^=.2n  —  2Jcs, 

in  which  Jc  is  the  constant  of  resistance.     This  equation  must  be 
combined  with  the  equation 

If  h  ,"i  and  ^  ,"'  are  the  respective  multipliers  of  these  equations  in 


—  351  — 

the  method  of  variations,  the  hrach/jstoclwonc  is  defined  h/j  the  differential 
equations 

f'i  =  ^, 

and  ])y  the  following  expression  of  the  normal  pressure  directed  in 
the  opposite  way  to  the  centrifugal  force 

Z>^  X2  sin  V  -\~  D^S2  cos  v  =.  - — . 

When  /c  vanishes,  the  value  of  ^t  is 

1 

and,  therefore,  the  value  of  ii  is  the  negative  of  the  reciprocal  of  the  ex- 
jpression  which  is  oUained  for  v  ivhcn  there  is  no  resistine/  medium,  and 
ivhich  is  independent  of  the  magnitude  of  the  fixed  force. 

597.      When  the  force  is  directed  toivards  a  fixed  centre,  the  nota- 
tion of  §  569  gives  by  (ooOjg)  for  the  value  of /x, 


598.  When  the  forces  are  parallel,  the   equation    (oolgi)    gives 
II  in  the  form 

'  COS  V 

599.  From  the  preceding  ecpiations,  the  crpiation  of  the  hrachgs- 
tochrone  of  a  heavy  hody  in  a  medium  of  constant  resistance  has  the  form 

R  sin  V 
^  ~  fl— /icos(»'  — ro)J ' 

in  wdiich  R,  h,  and  Vq  are  arbitrary  constants. 


—  352  — 

600.  In  a  medimn  of  ivhich  the  Imv  of  resistance  is  expressed 
as  a  given  function  of  the  velocity^  the  derivative  equation  of  mo- 
tion is 

in  which  F  is  a  given  function  of  v.  The  differential  equations,  by 
which  the  brachystochrone  is  defined,  become,  if  I  jU,  and  ji/j  are  the 
multipHers  of  (35O29)  and  (3524), 

D,  [ji  sin  v)  =  D^  £1 D,  ^1 , 
—  D,  {^  sin  v)  =  nj2  n.pi, 

The  reduction  of  these  equations  gives 

and  the  expression  of  the  normal  pressure  to  the  brachystochrone 
becomes 

D^  12  sm  V  -\-D^ll  cos  v  =  -^—  =  —^r-. =  —^ — rri^ — 

fix  Vv^  -|~  ^^ 

QH-^v^D^  V — q' 

601.  When  the  forces  are  parallel  to  the  axis  of  z,  the  equations 
(3529)  and  (352i7)  give 

a 

^        sinv' 

ir  a  \ 

'         •  sm  V        V 


—  353  — 

G02,     These    equations   give    for  the  h-achjstoclirone  of  a  heavy 
hodj  ill  a  rcsistiuf/  medium, 

by  wliicli  V  is  determined  in  terms  of  v.  The  substitution  of  this 
value  of  V  in  the  equation 

t*  D^.  V  j^ 

- — —  =  —  a  cos )'  —  y, 

gives  the  equation  of  the  hrachystochroiie  in  terms  of  q  awl  r.  The  pre- 
ceding formula}  include  the  results  obtained  by  Jellett  in  his  inves- 
tigation of  this  particular  case. 

When  V  is  inversely  proportional  to  the  velocity,  the  equation 
of  the  brachystochrone  may  assume  the  form 

2  A  [A  cos  2  (v  —  a)-^kjsm  2(v  —  a) 


Q  = 


m 


-j-  g  cos  r  [/(  cos  2  (r  —  «)  -h  ^'J 


When  V  is  proportional  to  the  square  of  the  velocity  and  has 
the  form 

the  equation  of  the  brachystochrone  is  derived  from  the  elimination 
of  V  between  the  equations 


cos(v  —  a)  = 


g  cos  a        f/  fin  r 


cos «    /gcosv         j\         [i')qco>a        o  /  \l/.7Cosr    ,     ,\ 

k7i;i^X—' ''■)  =  [-^' ocos(v-a)J(-^  +  /-). 

G03.  In  these  cases  of  the  brachystochrone  in  a  resisting 
medium,  it  is  apparent  that  the  condition  (3296)  is  usually  violated, 
and  that  Euler,  consequently,  erred  in  extending  this  proposition  to 
the  case  of  the  resisting  medium. 

40 


—  354  — 

604.  The  determination  of  the  form  of  the  curve  constitutes 
the  principal  feature  of  the  general  problem  of  the  brachvstochrone. 
But  the  nature  of  the  curve  may  be  given,  and  the  problem  is  then 
reduced  to  one  of  maxima  and  minima,  in  which  the  various  param- 
eters of  the  curve  are  to  be  determined.  Euler  has  shown  that 
there  is  a  peculiar  anah^tic  difficulty  in  some  problems  of  this  class. 
A  single  example  will  illustrate  this  species  of  inquiry. 

Let  the  given  curve  be  the  circumference  of  a  circle,  of  which 
the  plane  is  vertical,  and  let  the  ball  start  from  a  state  of  rest  at  the 
upper  point.  If, .then,  2  a  is  the  angle  which  the  line,  joining  the 
two  points,  makes  with  the  horizontal  line,  and  if  2  i  is  the  angle 
which  the  radius  drawn  to  the  upper  point  makes  with  the  vertical, 
the  equation  for  determining  i  is 


sec 


/  [f  ^  ( 1  jt)  —  I,  ( 2  «  —  /)]  —  [cot  2(i  —  a)-\-  cos  /] 


l-^,(in)-7f,{2a-i)]  +  '-^^ 


cos  2  a     I  sin  2  («" —  a) 


0. 


sin  z  « 


THE    nOLOCIIKOXE. 


605.  A  curve,  in  which  the  time  of  descent  along  a  given  arc, 
is  a  given  function  of  the  arc,  or  of  its  defining  elements  may  be 
called  a  holoehrone. 

006.  The  problem  of  the  holoehrone  becomes  simple,  tvhen  the 
forces  are  fixed,  and  the  tunc  of  descent  is  irroportional  to  a  given  'power  of 
the  arc.     Thus,  if  the  time  of  descent  is  expressed  by 

T,=  As'\ 

in  which  s  is  the  leng-th  of  the  arc.     Let 

±(1 


—  355  — 

in  wliijli  the  upper  sign  corresponds  to  tlie  case,  in  wliicb  n  is  less 
than  unity,  and  the  lower  to  that  in  \Yhich  n  exceeds  luiity.  The 
force  alonti;  the  curve  is 

When 

the  force  along  the  curve  is 

2  A.,s  s  ' 

GOT.  When  the  force  is  that  of  gravity,  the  equation  of  the 
holochrone  of  the  preceding  problem  assumes  the  form 

g  sin  T  r=:  —  B  b^~'~'\ 

608.  If  the  iime  of  descent  admits  of  heing  devetoped  according  to 
integral  ascending  poivers  of  s,  the  developed  expressions  of  S  and 
12^  are  obtained  from  the  formulce 

in  which  the  successive  terms  of  P^  are  obtained  from  the  equations 
represented  by 


W2L,=X['=-9r:fc)]- 


1 


0 

The  second  member  of  this  equation  is  to  be  developed  in 
form  precisely  as  if  f  were  the  symbol  of  derivation,  and  in  the 
result  there  must  be  substituted  for  P,  =  o  ^"^^^  f^''^  J-\=.q,  the  values 


—  356  — 

609.  W/mi  the  forces  are  fixed,  and  the  time  of  descent  is  a 
given  function  of  the  iniiial  value  of  the  potential,  the  problem  of  the 
holochrone  can  be  solved  by  the  method  applied  by  Abel  to  the 
case  of  a  heavy  body.  If  A  is  the  final  value  of  the  potential,  in 
which  the  arbitrary  constant  is  determined  so  that  the  potential 
may  vanish  with  the  velocity,  the  time  of  transit  expressed  as  a 
function  of  A,  assumes  the  form 

T — i-  r    ^^' 

■^^-'  s|•^J^^J{A  —  ^y 
T\\Q  integral,  relatively  to  A  of  the  product  of  this  expression, 
multiplied  by 


nsl{il—Ay 
is 


0 

But  the  notation 


rh=J^{—\ogxY-\ 

0 

with  the  familiar  equation 

1 

J^  x«-^   ra  r(i— k) 

0 

gives,  by  a  ready  reduction 

rv       1  C  _^^LZ!LJ\  —  —  rnril ;A—    ^^° 


;Note.  — The  notation  (SoG^o)  is  substituted  for  that  of  (91,,4),  which  was  unwisely 
introduced  instead  of  the  usual  form,  which  is  here  restored. 


€)■)( 


If  the  product  of  tins  equation  multiplied  by  cc  ff  («)  is  in- 
tegrated relatively  to  a,  and  if  the  function  /^  of  x  is  defined  by 
the  equation 

r(9(«)2-)=/„ 

t/  a 

so  that 

U  a 

the  integral  ffives 


0  0 

which,  when 
gives  by  (ooSie) 

n  Ja  \/{Si  — 


S —  * 


0 


The  general  relations  between  s  and  12  complete  the  solution, 
and  indicate  the  form  of  coordinates  in  which  the  solution  should 
be  finally  exhibited, 

610.  If  the  forces  are  parallel  to  the  axis  of  z,  12  is  a  function 
of  s,  and  the  elimination  of  z  between  (ooTis)  ^'^^'^  the  equation 

cos^  =  A-5, 

gives  this  holochrone  expressed  in  terms  of  the  length  and  direction 
of  the  arc. 

Gil.  If  the  forces  are  directed  towards  a  fixed  point,  which  is 
assumed  to  be  the  origin  of  coordinates,  the  elimination  of  ;•  be- 
tween  (SSTis)  ^^^ 

cos '  =  D,s, 

gives  this  holochrone  expressed  in  terms  of  the  length  of  the  arc 
and  its  inclination  to  the  radius  vector. 


—  358  — 
G12.     If  T^,  developed  according  to  powers  of  A,  is  expressed 

by 

it  is  evident  that 


-n/^-4^;^>-] 


613.  An  interesting  case  of  this  potcniial  lioloclirone  is  olttained, 
ivlien  the  hodij  is  supposed  to  approach  the  point  of  maximum  potential 
along  a  given  curve,  and  the  required  curve  is  to  he  such  that  the  whole 
time  of  oscillation  shall  he  a  given  function  of  the  maximum  potential.  If 
5i  denotes  the  given  arc,  the  time  of  oscillation  has  the  form 

rp   _J_    f     D^  (^  +  ^0  . 

so  that,  by  the  process  of  §  G09, 

u 

In  order  that  the  two  curves  may  be  continuous,  the  direction 
of  the  given  curve  must  coincide  with  that  of  the  level  surface  at 
the  point  of  maximum  potential.  But  this  direction  may  be  given 
by  an  infinitesimal  bend  at  the  extremity  of  the  curve,  so  that  this 
is  not  a  practical  limitation  of  the  problem. 

614.  If  the  given  time  of  oscillation  is  constant,  the  equation 
(oSSig)  assumes  the  form 


B{s  +  s,) 


2 —  r) 


f 


and  the  compouml  curve  hccomes  a  peculiar  species  of  tantochrone,  which 
was  investigated  by  Euler  in  the  case  of  heavy  bodies. 

615.      When  the  forces  are  not  ivhollg  fixed   hut   mag  depend  upon 


—  359  — 

the   vehcitf/,   the   'problem  of  the  holochrone   hccomes,  to   a   certain  extent, 
indeterminate.     For,  if 

TF=zO, 

is  an  assumed  equation  between  5,  t  and  v,  such  that  t  and  s  vanish 
together,  but  when  v  vanishes,  the  resulting  equation  between  6-  and 
t  assumes  a  given  form  corresponding  to  the  given  condition  of  the 
holochrone,  the  derivative  of  this  equation  gives,  for  the  expression  of  the 
force  along  the  curve, 


R 


from  which  the  time  is  to  be  eliminated  by  means  of  the  assumed  equation. 
61 G.     In   most  problems,  in  which  the    forces    are    dependent 
upon  the  velocity,  the  form  of  R  is  not  unlimited,  but  is  usually  so  re- 
stricted that 

in  ti'hich  Eg  is  a  function  of  s  and  represents  the  action  of  the  fixed 
forces,  tuhile  E^,  is  a  function  of  v  and  represents  the  resistances,  to  which 
the  body  is  suhjcct.  In  this  form  of  the  jDroblem,  geometers  have  not 
made  much  progress  towards  its  solution,  although  the  case  of  the 
tautochrone,  exhibited  in  this  aspect,  has  been  the  occasion  of  much 
discussion  and  many  difficult  memoirs. 

617.     If  the  equation  (SSOg)  solved   with    reference    to  /,  ac- 
quires the  form 

the  expression  for  JR  is 

l  —  v  n.  T. ,. 


B 


-D..  Z. 


Avliich   is  essentially   identical  with    L.vciRAMiE's    most  gn/cral  foj-mula 
in  the  case  of  the  tautochrone. 


—  360  — 

618.  If  the  equation  (SSOg),  solved  with  reference  to  v,   ac- 
quires the  form 

the  expression  for  R  is 

Avhich   formula  comprises  Laplace's  general  form  of  solving  the  iauto- 
chrone. 

619.  If  the  equation  (SSOg),  solved  with   reference  to  s,  ac- 
quires the  form 

the  expression  for  R  is 

p V  —  A  S,.,  t 

620.  When  the  equation  (SSOg)  is  presented  in  the  form 

in  which  T,  >S',  and  V  are  respectively  functions  of  ^,  s,  and  v,  the 
A'alue  of  7t  is 


R  =  — 


A  V 


But  D^T  h  a  function  of  t  and,  therefore,  of  S  -\-  V;  it  may, 
indeed,  be  any  arbitrar}^  function  of  >S'  -|-  V,  so  that  if  ijj  denotes 
this  arbitrary  function,  R  becomes 

621.  When,  in  the  preceding  section,  aS'  is  changed  into 
—  log  aS'  and 

V=  log  2?, 


—  3t)l  — 
the  value  of  H  may  be  presented  in  the  form 

which  is  the  same  with  a  familiar  formula  of  Lagrange  for  the  case 
of  the  iaidochrone. 

622.     The  cases,  in  which  the  formula  (SGlg)  assumes  the  form 
(359i5)  ^I'G  easily  investigated.     For  this  purpose  let 

V 

and  the  derivatives  of  (oGlg)  give 

7>„  7?  =  A  /  +  2 1'  A  log  S^D,.  i?,, 

D,D,R  =  —^^Dli^  2z  SDs  A  log  S=  0  ; 

whence 

i)2;^  =  2>S''AAlog>S'=2«, 

in  which  a  is  any  constant.     Hence 

7,  =  az'^-\^lz-\-c, 

in  which  h    and   e   are   constants  introduced    by  integration.     The 
value  of  R  is,  then, 

R^c,SJ^hv  +  (a  +  A 'S^) I' ; 
so  that,  if /^  and  II  arc  constants,  the  final  values  of  S  and  R  are 

R^eS  +  hv  +  hi^-, 
46 


OUJ    — 

and  this  formula  of  Lagrange  is  restricted  to  the  resisting  mediimi,  in 
vjJiich  the  resistance  has  the  form 

a  -\-1j  V  -\-  h  v^ 

which  was  first  remarked  by  Fontaine. 

The  form  of  T,  in  this  case,  may  be  derived  from  the  equations 

T 

c    '  —  z, 
—  I),  T=^  =  az  +  lj  +  '-=t>  +  ec''+ac-'^; 
which  trive 

2 e av S -\-  be S' -\-h a  ir 

bv  S-\-eS'^^av' 
y^h.^ilLl  V  vanishes  this  equation  becomes 

sJ[1ea)^Q^\{r.—t)s){;iea  —  t?)'\  =  h, 

so  that  the  interval  r  —  /is  independent  of  the  length  of  the  arc, 
and  the  curve  is  a  tautochrone  if  %  is  also  independent  of  s,  which 
is  the  case  when  S  vanishes  with  6',  that  is,  when 

h 

This  condition   is  always  observed,  if  the  direction   of  the    curve 
coincides  witli  that  of  the  level  surface  at  its  termination,  so  that  in 
every  case,  this  holochrone  is  cssentiaHij  tautochronous. 
623.     If,  instead  of  (3592.r,)  we  suppose 

and  if  (/'  denotes  an  arbitrary  function,  the  value  of  11  has  the  form 

l^'l}     ■*■  -«.  V 


OP  o 

OUO    

When 

in  "svhicli  aS' and  iS\  are  functions  of  s,  and  V  is  a  function  of  v,  the 
valne  of  II  becomes 

*  =  ST-  [''■  f-^'  ^"+  *■)  - '•  (*'  f'+  '^' )] ' 

which  inchides  Lagrange's  formula.  Forms  of  this  kind  may  be 
indefinitely  multiplied,  without  diminishing  the  difhculty  of  obtain- 
ing such  as  are  new  and  not  included  in  the  iuvestigations  of  §  622. 
624.  A  curious  case  of  the  holochrone  is  introduced,  when  the 
form  of  H  is 

in  which  S\  is  a  function  of  s.  The  only  case  of  (36I3),  which  can 
assume  this  form  is  easily  proved  to  be  that  of  (361gi)  when  jS 
is  left  undetermined.  If,  then,  the  factor  of  r^,  diminished  by  a 
constant,  is  inversely  proportional  to  the  radius  of  curvature,  t/ie 
form  of  the  t^esisiance,  by  including  in  it  part  of  the  term  e  S,  is 
that  of  (0623)  increased  !)>/  a  term  'proportional  to  the  friction  upon  the 
curve. 

If  the  fixed  force,  in  this  case,  is  that  of  gravity,  and  the  axis 
of  s  is  vertical,  and  if  v  is  the  inclination  of  the  radius  of  curvature 
to  the  axis  of  z,  the  first  and  last  terms  of  R  give,  if  k  is  the  con- 
stant of  friction, 

^  q  9^\x\v  -\-1c  g  cos  v 

O  =  — ' , 

e 

a  +  D,  aS'=  a  —  - =  —  ( A  4-  -  )  >S , 

1         a e  —  hg^mv  —  hhg cos  v  ^ 
Q  (1  —  F)  g  cos  V  ' 


—  3G4  — 

so  that  the  curve    determined  hi/  (oGlgo)  isjncluded  in  this  form.     This 
is  a  generalizi 
to  the  cycloid. 


is  a  generalization  of  Bertrand's  similar  investigation  with  regard 


THE     TACIIYTROPE. 


625.  A  curve  in  which  the  law  of  the  velocity  is  given  may 
be  called  a  tach/jtropc. 

626.  When  the  law  of  the  velocity  is  given  in  an  equation  Jjctiveen 
the  velocity^  the  space.,  and  the  time,  the  formulcv  o/*  §615  are  directly 
applicable  to  the  complete  solution  of  tJie  prohlem  ;  and  all  the  subsequent 
transformations  of  these  formulce  may  be  applied  to  the  present  case. 

627.  When  the  time  is  not  involved  in  the  equation  (oSQs), 
but  the  portion  R^  of  the  force  R  is  given,  the  other  portion  R,  is 
determined  by  the  equation 

*«  JJ     ]Y  ^''5 

from  which  v  is  to  be  eliminated  by  the  given  equation  (ooOgj). 
EuLER  has  solved  various  cases  of  this  tachytrope, 

628.  One  of  the  simple  examples,  solved  by  Euler,  is  when, 
in  the  case  of  a  heavy  body, 

^„  =  — /t?''% 

and  the  velocity  is  to  depend  upon  the  arc  in  the  same  form  as 
if  the  body  descended  in  a  vacuum  upon  an  inclined  straight  line, 
so  that  the  equation  (3593)  iicquires  the  form 

V^  :=.  hs, 

whence 

y  sin  V  =  R^=z^h-}- k  (h s)'"". 


3G5  — 


When 

this  equation  becomes 


m 


g  sin  V  ^=1  ^  h  -\-  Jc  hs, 


or  the  required  tachjtrope  is  a  cycloid. 

G29.  Another  simple  and  interesting  example  of  this  prol^lem 
was  proposed  by  Klingstierna  and  solved  by  Clairaut.  It  is  that 
of  a  heavy  body  in  a  medium,  of  which  the  resistance  is  propor- 
tional to  the  square  of  the  velocity,  approaching  the  origin  with  a 
velocity  equal  to  that  which  it  would  have  acquired  by  falling 
in  the  same  medium  through  a  height  equal  to  the  distance  of 
the  body  from  the  origin  measured  upon  the  curve.     In  this  case   - 

whence  the  equation  of  the  tachytrope  is 
of  which  the  integral  is 

630.  A  simple  example  of  the  problem  of  §  627  is  that  in 
which  the  velocity  is  uniforui.     In  this  case 

7?,  =:  —  B,  =  a  constant  =  A  -- , 

so  that  ill  the  case  of  a  licavij  hodij  this  tachjtrope  is  a  straight  line  ; 
in  tlmt  of  a  constant  force  directed  towards  a  fixed  point,  it  is  a  loga- 
rithmic spiral ;  and  in  every  case  the  sine  of  the  angle,  at  which  it  inter- 
sects each  level  surface,  is  inversehj  proportional  to  the  fixed  force  u'hich 
acts  at  the  iioint  of  intersecticn. 


—  366  — 

631.  When  the  given  forces  are  parallel  to  the  axis  of  ^,  and 
the  given  equation  (ooOg)  is  expressed  in  terms  of  v  and  0,  the 
equation  of  the  tachytrope  is 

{D,  n  sin  V  -\-  B,)  D,.  W-\-  A  Wv  sin  v  =  0, 

from  which  v  is  eliminated  by  means  of  the  given  equation.     Euler 
has  solved  several  cases  of  this  tachytrope. 

632,  If,  in  this  case,  the  curve  is  to  be  such,  that  the  velocity 
shall  have  a  constant  ratio  to  that  which  it  would  have  acquired  in 
a  vacuum,  the  equation  (3665)  assumes  the  form 

D,  12  sin  V  z=z  — 


1  — « 


If  the  resistance  is  proportional  to  the  square  of  the  velocity, 
so  that  li,.  has  the  form 

the  equation  of  the  tachytrope  is 

sin  V  A  log  (/2  -f-  //)  =  ^. 

633.  When  ihe  (jivcn  forces  are  directed  towards  the  origin,  and 
the  given  equation  (3593)  is  expressed  in  terms  of  v  and  r,  the  equation 
of  the  tachgirope,  in  a  medium  of  given  resistance  is 

{D,  n  cos :  +  i?„)  A-  W-\-  D,  Wv  COS :  =  0 

from  which  v  is  eliminated  by  means  of  the  given  equation. 

634.  If,  in  this  case,  the  curve  is  to  be  such  that  the  velocity 
shall  have  a  constant  ratio  to  that  which  it  would  have  acquired  in 
a  vacuum,  the  equation  (36624)  assumes  the  form 

A-^^cos;  =  — -^. 
1  — a 


—  867  — 

If  the   resistance  has  the  form    (36Giq),    the    equation    of  tlie 
tachytrope  is 

2  a  k 


cos:  A  log  (/>  +  //) 


I— a' 


Goo.  W/icii  the  laiv  of  the  vclocibj,  in  a  medhun  of  Jrnown  rcsid- 
ance,  is  given  in  a  given  direction,  such  for  instance  as  that  of  the  axis 
of  X,  and  so  given  that 

t'cos^=  W,^^, 

in  Avhich  W,  j.  is  a  given  function  of  s  and  x,  the  equation  of  the 
tachytrojDe  is  derived  from  the  equation 

{D,  n  -]-  li^)  cos  ^  —  ^^  sin  ^  =  ^'  A  W,^  ^  +  v  cos  ^  D^  W,_  ^ ; 

from  Avhich  v  is  eUminated  by  the  given  equation. 

036.  When  the  velocity  in  the  given  direction  is  uniform, 
these  equations  become 

V  cos  %  =  a , 

a-  sin  ^ 

637.  When  the  given  force  is  that  of  gravitg,  and  (i  is  the  in- 
clination of  the  given  line  to  the   vertical,  the  eqliatiou  of  this  tachytrope 

hcco)ncs 


(5rcos(p,'  — ;:)  +  i?,)  cos'' 


This  problem  is  solved  by  Euler  in  the  case  in  which  the  given 
direction  is  horizontal  and  in  that  in  which  it  is  vertical.  A  special 
solution  is  obtained  upon  the  hypothesis  of  a  constant  velocity ;  in 
this  case,  the  tachytrope  is  a  straight  line  determined  by  the  con- 
dition 

^cosGi  — :;)  +  i?„  =  o. 


—  368  — 

638.      When  there   is   no   resisting  medium,  the  equation  (oGTsi)  ({f 
the  tachytrope  becomes 


^         g  cos^  tc  cos  {^  —  ly 

When  the  hne  is  horizontal 
and  the  equation  becomes 

a" 

^         g  cos^  % 

so  that  the  tachjtrope  of  this  case  is  a  iiarahola. 
When  the  line  is  vertical 

/^  =  0, 
and  the  equation  becomes 


a  sin  \. 

(>  = n? 


SO  that  the  tachjtrope  of  this  case  is  the  evolutc  of  the  ijarahola. 
With  the  notation 


2  dr 

the  equation  (3683),  expressed  in  rectangular  coordinates  is 

11)\l{x-^y  cot  /i)  —  t)x  =  lQoi  (-i  log  [cot  l^>-\-b^{x  -\-y  cot  f:?)] . 
639.     If  the  resistance  is  proportional  to  the  velocity,  so  that 

and  if  the  direction  of  the  line  in  which  the  velocity  is  given  is 
such  that 

g  cos  /•>  ■=^ka, 

the  equation  of  the  tachytropc  of  a  heavy  J)ody  is 

X  sin  {i  — y  cos  [■)  z=z  —  c    a  , 


369 


THE     TACHYSTOTKOPE. 


640.  The  curve  on  which  the  final  velocity  in  a  given  resist- 
ing medium  is  a  maximum,  may  be  called  a  tachjdotropc. 

641.  In  a  medium  in  which  ihe  law  of  resistance  is  expressed  as 
it  is  in  §  600,  the  notation  of  that  section  gives  for  the  differential 
equations  of  the  tachystotrope 

D^  (u  sin  v)  =  D^  11  D,  //j, 
—  D,  (a  cos  J')  =  D,  £2  D,  //j , 
V  D,Ui  =  III  D^  V. 

The  reduction  of  these  equations  gives 

D,^i  =  n,S2I),^,i  =  I),{u,V), 

^  =  Hi  V, 

and  the  expression  of  the  normal  pressure  to  the  tachj^stotrope  be- 
comes 

V 


cAiogK' 

642.     In   the   case   in  which  the  law  of  the   resistance  is  ex- 
pressed by  the  formula 

the  normal  pressure  becomes 

D  12  =  — 

P  in  () ' 

so  that  the  normal  press^ire  has  a  constant  ratio  to  the  centrifugal  force, 
wdiicli  result  was  obtained  by  Euler  in  the  case  of  a  heavy 
body. 

47 


—  370  — 

643.  When  the  resistance  is  constant,  the  tachjstotrope  is  a  straight 
line. 

644.  When  the  forces   are  imrallcl  to  the  axis  of  z,   the    equa- 
tions (3699)  and  (oSDie)  give 


fi  =  u^  V=  - 


a 

sin  >'' 


645.  The    equation   of  the   tachjstotrope    of  a  heavy    hoJij   is   ob- 
tained, therefore,  Inj  the  elimination  of  v  hetivccn  the  equations 

V= ^" 

b  sin  V  —  a  cos  v ' 

V  D^,  V  a  a 

—  =  —  q  COS  V  —  -j—. — ^^ . 

(>  "^  b?,mv  —  acosv 

646.  When  V  has   the  form  (3692,i),   the  equation  of  the  tachy- 
stotrope  of  a  heavy  hody  is 

a 


-J—. -T,  =  Jr-  (m  a  o  sin  v\ 

{b^va.v  —  acosi')'  \     t/  >>  j 


THE  BARYTROPE  AND  THE  TAUTOBARYD. 

647.  The  curve,  in  which  the  law  of  pressure  is  given,  may 
be  called  a  harytrope,  and  that  bar^^trope,  in  which  the  pressure  is 
everywhere  the  same,  may  be  called  a  tautoharyd. 

648.  When  the  pressure  is  a  given  function  of  the  arc,  which 
may  be  denoted  by  S,  its  equivalent  expression,  if  F  is  the  fixed 
force  which  acts  in  the  direction  /,  is 

-  — i^sin{==>S'; 

and  the  differential  equation  of  the  harytrope  is 

2R  =  D,  [*)  (^'4- i^cos,0]  =  2  ^sin;;+  2  E,. 


—  ot  L  — 

649.  Iji  ilie  case  of  a  hcavn  hodf/,  if  the  axis  of  z  is  vertical,  the 
differential  equation  of  this  havjjtrope  hecoincs 

from  which  v  may  be  ehminated  by  means  of  the  equation 

^'^  :=!  ct  [S -\- g  sin  v) . 
In  this  case,  the  differential  ecjiiation  of  the  tautoharyd  is 
(a  -{-(/  cos  v)  D,  0  =  3^  sin  v  -\-2B^. 

650.  ^yhcn  the  resistance  is  constant,  the  equation  of  the  harf/trope 
of  §  G48  is 

i>  {S  +  Fco.p  =  2  [n  +  //)  +2  .7?„. 
In  the  case  of  the  heavy  lody,  this  equation  becomes 

^^■i?,5+^cosi/7?,.  =  2y.-  +  2//+257?„j 
and  that  of  the  taidol}aryd  is 

[a  -\-g  cos  vj"^^  ()  =.  A  \_g  -f-  a  cos  v  -]-  sin  v  \J  (y^  —  ^^^'YW 
if  A  is  an  arbitrary  constant, 

and 


But  if 

the  equation  of  the  tautohargd  is 

lo^'  K  ( ^^  +  y  COS  V  Y   =  -y—i ^  cos^    ^J  ~ . 


O'-Q 


When  there  is  no  resistance,  the  tautoJ)ar>/d  of  the  heavy  hocly  is 
defined  hij  the  equation 

_  A 

651.  When  aS'  vanishes,  there  is  no  pressure  against  the  bary- 
trope,  and  this  curve  is  that  on  which  the  body  moves  freely. 
Thus  the  equation  of  the  barjtrope  of  the  heavy  body  becomes, 
under  this  condition, 

A 


^^  ~  (cos  rf  ' 

tvMch  is  that  of  a  parabola. 

652.  When  the  curve  of  the  harytrope  is  f/iven,  the  eqiicdions  (37O27) 
and  (STOgi),  determine  the  laiv  of  the  fixed  force  when  that  of  the  re- 
sistance is  hioivn,  or,  reciprocally ,  that  of  the  resistance,  lehen  the  fixed 
force  is  hnown. 

653.  When  the  forces  are  parallel  to  the  axis  of  z,  the  equation 
(37O31)  becomes 

*  \     s  /    I  COS- ;' 

lohich  is  appliccd)le  ivhen  the  curve  is  given. 

When  there  is  no  resistance,  this  equation  gives 

Fo  cos-^  V  =  — /  [cos^  V  n,  {S'q)-]  =  — ^  [cos^ )/  n^  {S())'] . 

654.  In  the  case  of  pa?rdlel  forces,  when    the  tautobaryd  is  a 
circle,  and  there  is  no  resistance,  the  fixed  force  has  the  form 


F=: 


Q COS' V 

m  which  b  and  F  must  vanish,  if  v  can  become  a  right  angle. 


When  the  fixed  force  is  that  of  gravity,  and  the  tantobaryd 
is  a  circle,  the  expression  of  the  resistance  is 


J!.  =  -lff{^-a) 


655.  In  the  case  of  jmrallel  forces,  when  the  tantobaryd  is  a 
cycloid  of  which  the  base  makes  an  ano-le  a  with  the  direction  of 
the  parallel  forces,  and  when  there  is  no  resistance,  the  equation  of 
the  cycloid  being 

^  r=2  ^  sin  (j^  —  a), 

the  expression  of  the  force  is 

^        «sin(r  —  «)  4- ^  a  sin  (3  *  —  «) -j- ^  a  sin  (r -]- «) -|- 5 
2  sin  (r  —  a)  cos' v 

When  h  vanishes  and  a  is  a  right  angle,  this  expression  is  re- 
duced to 

F=^  I  a  coseci', 

which  coincides  with  Euler's  solution  of  this  example. 


THE    STNCnROXE. 

656.  The  surface  or  curve  which  is  the  locus,  at  any  instant, 
of  all  the  bodies  which  start  simultaneously  from  a  given  point 
w^ith  a  given  velocity,  and  move  upon  paths  which  are  related  by 
a  given  law,  is  called  a  si/nchrone,  and  the  given  starting  point  may 
be  called  its  dynamic  pole.  This  class  of  loci  was  first  discussed  by 
John  Bernoulli. 

657.  If  an  integral  of  the  motion  of  the  body  along  one  of 
the  paths  to  the  synchrone  is  obtained  in  the  form 

TF=0, 


—  374  — 

in  which  W  is  a  function  of  the  time,  of  the  arc  of  the  path,  and  of 
the  parameters  by  which  the  relationship  of  the  paths  is  expressed ; 
this  equation  is  the  required  equation  of  the  synclirone^  if  the  time  is  as- 
sumed to  be  constant ;  and  it  is  referred  to  the  system  of  coordinates,  con- 
sisting of  the  described  arc  and  the  given  parameters. 

658.  If  the  only  force  is  that  of  a  resisting  medium,  and  if 
the  form  of  the  path  is  given,  and  also  the  position  of  the  dynamic 
pole  upon  it,  but  not  its  direction  in  space,  the  synchrone  is  obviously 
the  surface  of  a  sphere,  of  ivhich  the  dynamic  pole  is  the  centre. 

659.  If  the  body  moves,  without  external  force  and  without 
resistance,  upon  a  straight  line,  which  rotates  uniformly  about  a 
given  axis  passing  through  the  dynamic  pole,  the  synchrone  is  a 
surface  of  revolution  about  the  same  axis,  and  it  is  defined  by  the  polar 
equation  (SSOso)  or  (25I3)  ivlien  p  vanishes  and  t  is  constant. 

660.  When  the  fixed  forces  arc  directed  toivards  a  point,  or  ivhen 
they  are  p)arallel,  the  synchrone  of  bodies  moving  upon  straight  lines,  is  a 
surface  of  revolution,  of  which  the  axis  is  the  line  of  action  tuhich  passes 
through  the  dynamic  pole. 

661.  In  the  rectilinear  motion  of  a  heavy  body,  it  is  obvious  from 
(255i3),  that  the  2^olar  equation  of  the  synchrone  has  the  form 

r  =  a  cos  l-\-b, 

ivhich  becomes  a  sphere,  ivhen  b  vanishes,  that  is,  ivhen  the  initial  velocity 
vanishes. 

662.  In  the  rectilinear  motion  of  a  heavy  body  through  a  medium, 
of  ivhich  the  resistance  is  proportional  to  the  square  of  the  velocity,  the 
polar  equation  of  the  synchrone  has  the  form, 

Ac''=Cos{Bcos^:). 


—  375  — 


THE    SYXTACIITD, 


GG3.  The  surface  or  curve  ^vliicli  is  the  locus  of  all  the  poiuts, 
at  which  bodies  have  the  same  velocity,  when  they  move  from  a 
given  point,  Avith  a  given  velocity,  upon  paths  which  are  related  by 
a  given  law,  may  be  called  a  Sf/ntachi/d. 

664.  If  an  integral  of  the  motion  of  a  body  along  one  of  the 
paths  which  proceed  to  the  syntachj'd  is  obtained  in  the  form 

W=  0, 

in  which  W  is  a  function  of  the  velocity,  of  the  described  arc,  and 
of  the  parameters,  this  equation  is  that  of  the  syntachyd  in  the 
same  form  of  coordinates  with  those  in  which  the  sj^nchrone  of 
§  657  is  expressed. 

665.  In  the  case  of  §658,  the  syntachjd  coincides  ivitli  the  sijn- 
chrone. 

666.  In  the  cases  of  §§  659  and  660,  the  syntachyd  is  a  surface  of 
revolution  about  the  same  axis  iviih  the  synchrone. 

667.  When  the  action  is  exclusively  that  of  fixed  forces,  the  syn- 
tachyd is  a  level  surface. 

668.  ^Yhen  a  heavy  body  moves  upon  a  straight  line,  on  which  there 
is  a  constant  friction,  and  through  a  medium  of  ivhich  the  resistance  is 
proportional  to  the  square  of  the  velocity,  the  equation  of  the  syntachyd  is 

in  which  the  notation  of  §  5]  5  is  adopted,  A  and  B  are  constants 
and 

a  =  g  tan  a  . 

669.  When  a  heavy  body  moves  vpon  a  straight  line,  on  which 
the  friction  is  constant  and  through  a  medium   of  ivhich  the  resistance  is 


'proportional  to  the  vclocit/j,  the  cqiuition  of  the  srjntachjd  has  the  form 
log  iA  -  cos  (.  +  .)]  =  i?  -  ^^^-^^ . 

670.  When  the  hody  moves  upon  a  line  on  vhich  the  friction  is 
constant  and  through  a  medium  of  tvhich  the  resistance  is  proportional  to  the 
square  of  the  velocdi/,  the  equation  of  the  sf/ntachf/d,  expressed  in  the  form 
of  coordincdes  of  §  657,  is 

which  coincides  with  Jacobi's  investigation  of  this  case  of  motion. 


A   POINT    MOVING    UPON    A    FIXED    SURFACE. 

671.  Among  the  various  forms,  in  which  tlie  motion  of  a  point 
upon  a  fixed  surface,  with  fixed  forces,  can  be  discussed,  that  of  tlie 
principle  of  least  action  is  here  selected.  In  this  case,  therefore, 
the  whole  amount  of  action,  denoted  by 

is  to  be  a  minimum.     If,  then,  the  equation  of  the  surface  is 

X=:0, 

if  rectangular  coordinates  are  adopted,  if  pi  is  the  multiplier  of  the 
preceding  equation  of  the  surface,  and  ^a  that  of  the  conditional 
equation 

the  equation  of  the  path  of  the  body,  with  reference  to  either  axis,  is 
D.  V  +  a,  D,L  —  D,  ifi  .1-0  =  0 . 


—  377  — 

The  sum  of  these  three  equations,  multiplied  respectively  by  x\  y, 
and  /,  is 

or 

V  ^  a. 
Whence 

672.  If  the  tangent  plane  to  the  given  surface  is  assumed,  at 
each  instant,  to  be  that  of  .r//,  and  if  the  axis  of  jj  is  taken  normal 
to  the  path  of  the  body,  the  preceding  equation  becomes,  if  Oj  de- 
notes the  radius  of  curvature  of  the  projection  of  the  path  upon  the 
tangent  plane, 

—  D  11- 


so  that  the  ccntnfvgal  force  of  the  lody  in  the  direction  of  the  surface  to 
vMcli  it  is  restricted  is  equal  to  the  normal  irressiire  upon  the  path  in  the 
direction  of  the  tangent  plane. 

673.  ^Vhcn  the  direction  of  the  force  is  normal  to  the  surface,  tvhich 
is  the  case  ivith  the  level  surface,  or  tvhen  there  is  no  force,  the  path  of  the 
hody  is  the  shortest  line  tvhich  can  he  drawn  upon  the  surface,  and  coincides 
ivith  the  hrachystochrone. 

674.  When  the  velocity  is  constant,  the  erpiation  (377]3)  exj^resses  the 
condition  that  the  hody  may  move  upon  the  intersection  of  a  level  surface 
ivith  the  given  surface.  In  this  case  o^  is  the  radius  of  curvature  of  this 
intersection,  and  Dy  12  is  the  whole  force  in  the  direction  of  the 
tangent  plane  to  the  surface. 

675.  When  the  velocity  is  a  given  function  of  the  parameter  of 
the  level  surfoce,  the  equation  (377i3),  with  the  notation  of  the  pre- 
ceding section,  expresses  the  equation  of  a  surface  over  which  the 
body  moves  upon  the   intersection   of  this  surface   with  the   level 

surface, 

48 


—  37S  — 

676.  When  the  force  is  directed  toivards  the  origin^  and  the  given 
surface  is  a  plane  passing  through  the  axis,  the  equatiou  (oTTi.s),  com- 
bined with  (316i8),  gives  in  the  notation  of  §  569 

D,£l    2     2D,p 


SI  —  i(iu  (I  sill ',  p      ' 


of  which  the  integral  is 

(2 o  =  ^  z=  i  r  =  -^  Dt  s^. 

Whence,  if  (p  is  the  angle  which  r  makes  with  the  axis, 

ir'I),^,=pl 

But  i  t^dcf  is  the  elementary  area  described  by  the  radius  vector  in 
the  instant  di,  and  it,  therefore,  follows  that  the  area  described  lu  the 
radius  vector  is  proportional  to  the  time. 

The  equation  (STSu),  combined  with  that  of  living  forces,  gives 


i;,r  —  rv/  [(2  r-  {Si  —  Sl,)—4.pW  ' 

Whence 


^  ■"  '   rsJl2i^{U  —  Sl,)  —  lp\]' 


-I 


which  is  the  polar  equation  of  the  path  of  the  hodg.  That  this  equation 
can  be  obtained  by  integration  by  quadratures,  is  a  simple  case  of 
the  principle  of  the  last  multiplier. 

677.     When  the  potential  of  the  force  has  the  form 


i'-  =  -^, 


and  the  initial  velocity  is  such  that 

12  —  0 


—  379  — 
or 

l^  r=z  2  /2  =  — 

the  polar  equation  of  the  path  of  the  hodij  is 

plr"-'=\f{^a)  sin  [(;^  —  1 )  (9  —  «)], 

which  was  given  by  Riccati. 
K 

the  law  of  the  force  is  that  of  gravitation^  and  the  path  is  a  ^^wr^Jo/^  cf 
vMch  the  oricjin  is  the  focus. 
If 

the  attractive  force  is  inversely  proportional  to  the  cuhe  of  the  radius  vector, 
and  the  pcdh  is  a  logarithmic  spiral,  which  was  proved  by  Newtox. 
If 

the  attractive  force  is  inversely  proportional  to  the  fourth  power  of  the  radius 
vector^  and  the  path  is  the  epicycloid  formed  hy  the  exterior  rotation  of  a 
circle  upon  an  equal  circle,  which  was  proved  by  Stader. 

If 

n  =  2, 

the  attractive  force  is  inversely  p>roportional  to  the  fifth  iwwer  of  the  radius 
v^tor,  and  the  path  is  the  circumference  of  a  circle,  which  was  proved  by 
Newton. 
If 

the  attractive  force  is  inversely  proportional  to  the  sixth  power  of  the  radius 
vector,  and  the  path  may  he  called  the  trifolia  of  Stader,  by  whom  it  was 
investigated. 


—  380 
If 


n  =  o. 


the  attractive  force  is  invcrscl//  proportional  to  the  seventh  poivcr  of  the 
radius  vector,  and  the  path  is  the  Icmniscate  of  James  Bernoulli^  which 
was  proved  by  Stader. 

If  ' 

n=^  —  1 , 

the  repulsive  force  is  proportional  to  the   radius  vector,  and  the  fath  is 
an  equilateral  hjperltola. 
When 

r  becomes  infinite  when  (^  —  a)  vanishes,  which  was  remarked  by 
Stader. 

678.     When  the  vahies  of  £2,  S2q  and  ^h  ^i'G  such  that,  if  II  is 
an  integral  function  of  an  integral  root  of  r, 

B  =  s/{_2r^O-n,)-4.pr], 

the  expression  of  cp  in  (87822)  admits  of  integration.  For  if  the 
integral  root  of  r  is  denoted  by 

ri=  \Jr, 

and  if  the  notation  of  the  residual  calculus  is  adopted,  the  equa- 
tion (37822)  becomes 

mi  . 

log(Vr— ri) 


^  =  2»^'?i;;;^>=^»^'^i."&i 


679.     An  example  of  the  preceding  section  occurs,  when  m 
is  unity  and 

11  =z  a  r^  ~{-  ^  i'  -\-  c , 


—  381  — 
•which  corresponds  to 

n,=z  —  hV-  —  ac, 
and  an  attractive  force  of  the  form 

—  a"  r  —  ab-\---v,-\ -^^-^  . 

In  this  case  the  value  of  (p  is 

•^  e       ^R        e\J{A:ae  —  b')  sj  {Aae  —  b'y 

=  —  log;  ^  H , ,,.,      . — -  1  an^   ^J  -j-jr, — -, — r 

When  h  vanishes,  these  expressions  become 
n 1  ,,2  .2  _i  ^''  "Mt'i 

the  attractive  force  is 

and  the  equation  of  the  path  is 

«  +  ^=  -i;(f-«). 
When  e  vanishes,  the  expressions  become 

n  — 1 7.2 

the  repulsive  force  is 

2  1  7  ^P* 

a  r-\-  ao  —  — r , 


—  382  — 
and  tlie  equation  of  the  path  is 

When  1?  —  4  (^  e  vanishes,  the  equation  of  the  path  is 

:,  (ffi (Z)  = — -  —  W  ( 2  «  4-  -) . 

680.     Another  example  of  §678  occurs  when 

r,  =  ar  +  i+l, 
which  corresponds  to 

and  an  attractive  force  of  the  form 

«5    ,    &-  +  2 « e  +  4 joj    .    3  i e    ,    2  fi2 

The  equation  of  the  path  is 

^V(^^-4..)Tan[N^^^^-^-)(y-.)]. 
When  «  vanishes,  the  value  of  11^  vanishes,  the  attractive  force  is 

P-^ipi    .    She    .    2e^ 


^.i  ? 


and  the  equation  of  the  path  is 

hg{br  +  e)  =  ~{(p  —  a). 
When  P  —  iae  vanishes,  the  equation  of  the  path  is 


9 


ar~\-b=z 


a  —  fjp 


—  383  — 
681.     Another  example  of  §678  occurs  when 


in  which  case 


and  the  equation  of  the  path  is 
nB 


i(«  — 9)  =  log(l  +  ^r     "'), 


2  mp 

682.  TJie  forms,  in  vMch  (37822)  admits  of  explicit  integration 
iiitliout  any  special  determination  of  S2q  and  p,  are  included  in  the  f/cn- 
eral  expression 

in  v.'liich  h  is  two,  or  the  negative  of  unity,  so  that  SI  only  consists  of  two 

terms,  of  tvhich  one  is 

b 

and  the  general  fonn  of  the  central  force  consists,,  therefore,  of  two  terms 
of  ivhich  one  is  inversely  proportional  to  the  cube  of  the  radius  vector,  and 
the  other  may  he  either  xlirectly  propoiiional  to  the  radius  vector,  or  in- 
versely proportional  to  the  square  of  the  radius  vector. 

683.  In  general,  it  is  apparent  that  the  addition  of  a  term  to 
the  central  force,  which  is  inversely  proportional  to  the  cube  of  the 
radius  vector,  does  not  augment  the  difficulty  of  determining  the 
path  of  the  body.  In  any  ecpadion  of  a  path  of  a  hody  described 
under  the  action  of  central  forces,  which  is  expressed  by  the  elements 
(p  —  a,Y  and  t,and  tvhich  may  also  involve  the  constant  pi,  the  multi- 
plication of  the  angle  cp  —  a,  and  of  p^  by  the  factor 


B 


Ai-4)' 


—  384  — 

gives  the  equation  of  the  ixdh,  when  the  central  force  is  increased  hj  the 
term 

G84.  When  there  is  no  force  the  path  is  a  straight  line,  so  that 
ivhen  the  central  force  is  inversely  projwrtional  to  the  cube  of  the  radius 
vector,  the  polar  equation  of  the  path  is 


r  cos  [B  [ip  —  a)']-=:  B plJ 


If  the  force  is  repidsive,  B  exceeds  unity,  the  path  is  convex  to  the 
origin,  and  its  convexity  increases  with  the  increase  of  the  repulsive 
force  until  it  terminates  in  a  straight  line.  If  the  force  is  attractive, 
and  B"^  positive,  it  is  less  than  unity,  the  path  is  concave  to  the 
origin  but  of  infinite  extent,  and  the  concavity  increases  with  the 
increase  of  the  attractive  force  until  it  terminates  in  the  reciprocal 
spiral  of  Archimedes.  If  the  force  is  attractive,  B'^  negative  and 
I2q  positive,  the  equation  of  the  path  is 

r  Cos  \B  (c/,  _ «)  y/  -  1]  =  ^;>?  \/^, 

so  that  the  greatest  distance  of  the  path  from  the  origin  is  limited, 
and  the  path  is  a  spiral  about  the  origin  in  which  it  terminates,  at 
each  extremity,  through  infinitely  compressed  coils.  If  the  force  is 
attractive,  and  B""  and  /ig  negative,  the  equation  of  the  path  is 

r  Sin  {B  (r/)  _  a)  y'  —  1]  =  B p\  y/— , 

so  that  the  curve  extends  to  an  infinite  distance  from  the  orig-in  at 
one  extremity,  and  terminates  in  an  infinitely  condensed  coil  about 
the  origin  at  the  other  extremity.     In  these  three  cases,  the  formula 


—  385  — 
for  the  time  which  corresponds  to  (8849)  is 

t^  —  -^iVii\\_B  (ip  —  a)], 
the  formula  for  (3842o)  is 

and  that  for  (08427)  '^^ 

t  =  ?J^=^  Coi[_B  [cf  —  a)  si  —  r]. 

This  law  of  central  force  has  been  discussed  by  several  geometers, 
and,  with  peculiar  regard  to  the  special  cases  of  the  problem,  by 
Stader,  whose  results  coincide  substantially  with  those  of  this 
section. 

685.  When  the  central  force  is  ]j'^opo7iion(d  to  the  radius  vector, 
the  path  is  a  conic  section  of  ivhich  the  centre  is  at  the  origin.  It  is  an 
ellipse,  if  the  force  is  attractive,  and  an  hyperhola,  if  the  force  is  repidsive. 
In  the  case  of  the  ellipse,  if  a  point  w^ere  to  start  from  the  ex- 
tremity of  the  major  axis  at  the  same  instant  with  the  body,  and 
move  upon  the  circumference  of  which  this  axis  is  the  diameter, 
with  such  an  uniform  velocity  as  to  complete  its  circuit  synchro- 
nously Avith  the  body,  the  body  and  the  point  are  always  upon  a 
straight  line  which  is  perpendicular  to  the  major  axis.  For  dif- 
ferent ellipses,  the  time  of  description  is  proportional  to  the  square 
root  of  the  area.  In  the  case  of  the  hyperbola,  if  a  catenary  is 
drawn  through  the  extremity  of  the  transverse  axis,  in  such  a 
position  that  this  axis  is  the  direction  of  gravity,  while  its  ex- 
tremity is  the  lowest  point  of  the  catenary,  and  of  such  a  mag- 
nitude that  the  radius  of  curvature  of  the  catenary  at  this  point 
is  equal  to  the  semi-transverse  axis,  and  if  a  body  starts  upon  the 

49 


—  386  — 

catenary  simultaneously  with  the  given  body,  and  proceeds  in  such 
a  way  as  to  recede  uniformly  from  the  transverse  axis  with  a 
velocity  equal  to  that  of  the  given  body  at  its  nearest  approach 
to  the  origin,  the  line  which  joins  the  two  bodies  will  always  re- 
main perpendicular  to  the  transverse  axis  of  the  hyperbola. 

686.  W/icu  in  addition  to  the  term,  which  is  ^J^oporiional  to  the 
radius  vector,  the  central  force  has  a  term  inversely  'proportional  to  the 
cube  of  the  radius  vector,  the  path  can  he  derived  from  the  preceding 
section  hy  the  principle  of  §  683. 

When  the  term  which  is  proportional  to  the  radius  vector  is 
attractive  and  expressed  by 

a  r, 
the  polar  equation  of  the  curve  is 

'-^^  +  ^-^  =  v/[^^^^^o-4^^^^;4]cos[2i?(c^-«)] 

=  y/ [  02 _  4  «  i52^4]  Cos  [2  i?  (9)  —  c^)  v^  —  1] 
r=  v/ [4  « i?2;4  — '^-^a  Sin  [2  i? (9)  —  6.)  V/ —  1] . 

When  a  is  positive,  therefore,  the  path  does  not  extend  to  infinity, 
although  when  B^  is  negative  it  is  compressed  at  each  extremity 
into  an  infinite  coil.  But  when  a  is  negative,  the  term  propor- 
tional to  the  radius  vector  is  repulsive,  and  the  curve  extends  to 
infinity  if  B^  is  positive  ;  but  if  B^  is  negative  the  curve  is  limited 
if  lio  i*^  negative,  or  it  may  necessarily  extend  to  infinity  if  il^  is 
positive. 

In  the  special  case  of 

tan  (2  n  B71) p\    , 

Yb      —  H.v— «? 


-^0 


the  curve  is  asymptotic  to  itself 

687.      When  tlie  central  force  is  inversely  proportional  Jo  the  square 


—  387  — 

of  the  radius  vector  ivliich  is  the  law  of  gravitation,  the  yath  is  a  conic 
section,  of  ivhich  the  origin  is  the  focus.  When  the  force  is  attractive,  the 
imth  is  an  ellipse  if  il^  is  positive,  a  paralola  if  S2q  vanishes,  and  it  is 
that  branch  of  the  hgperhola  ivhich  contains  the  focus,  if  S2q  is  negative. 
Bui  when  the  force  is  repidsive,  the  path  is  that  branch  of  the  hgperbola 
ivhich  does  not  contain  the  focus.  The  farther  consideration  of  this  law 
of  force  is  reserved,  in  this  connection,  for  the   Celestial  mechanics. 

688.  When  in  addition  to  the  term,  ivhich  is  inversely  proportional 
to  the  square  of  the  radius  vector,  the  central  force  has  a  term  inversely 
proportional  to  the  cube  of  the  radius  vector,  the  path  can  be  derived 
from  the  preceding  section  bij  the  principle  of  §  683. 

If  the  term  of  central  force,  which  is  inversely  proportional 
to  the  square  of  the  radius  vector  is 


the  polar  equation  of  the  path  is 

iZ^'  _  «  ^  ^ («2 _  8  n^ B^pX)  cos  IB  (if -a)-] 

=  y/ (,,2 _ 3 X2^ B'^pt)  Cos  [B{if  —  a)sJ—l] 
==  v/(8  12,  B'-pt  —  d')  Sin  [B  (y  —  a)  sj  —  1], 

when  S2q  is  positive,  therefore,  the  curve  is  finite ;  it  returns  into 
itself  if  B^  is  positive,  but  if  B^  is  negative  it  terminates  at  each 
extremity  in  an  infinitely  compressed  coil  about  the  origin.  When 
S2q  is  negative,  one  portion  at  least  of  the  path  extends  to  an  in- 
finite distance  from  the  origin  ;  if,  moreover,  a  is  positive  and  & 
negative,  but  such  that 

a'>Sn,B'^pt, 
another  portion  of  the  path  is  finite  and  terminates  in  the  origin, 


—  388  — 

through  an  infinitely  compressed  coil,  while  the  two   infinite  por- 
tions commence  in  such  a  coil ;  if  the  negative  B"^  is  such  that 


a^<%n,B''p\ 


or  if  a  is  negative  as  well  as  £2^,  the  curve  only  consists  of  the 
portion  which  extends  from  the  coil  to  infinity.  The  time  may  be 
computed  by  the  three  formulae,  which  correspond  to  the  three 
forms  of  (387i8), 

S[^-^(^-'2.  +  ^(^._Ji2.),.sin[i.(,-.)])] 

=  —:;{87^l|P^  tan  [*.?(,-«)], 

r„ [^-^ (^ ^4  +  y/ (hI^.  +  i -^-'o)  r  Sin  [i?  (<;-«)  y/- 1])] 

S[^>(^^2.  +  ^(j^_iii„),.Cos[i?(<;-«)v/-l])] 
=       ^^_sc:i,B-^pt)     'CotliB{cp—a)^—l-]; 

the  upper  of  the  double  forms  of  the  first  member  applies  to  the 
case  in  which  S2q  is  positive,  and  the  lower  to  that  in  which  S2q  is 
negative.     This  case  was  partially  developed  by  Clairaut. 

689.  The  principle  of  §  683  may  be  extended  to  §  677,  and 
among  the  resulting  curves,  that  in  which  7i  is  2,  deserves  to  be 
noticed  from  its  simplicity,  the  equation  of  this  case  is 

690.  The  laiv  of  central  force,  for  tvJiich  ilie  integrals,  involved  in 
the  equations   of  motion,   can   he   expressed   hj   the  elliptic  forms  u'ithout 


—  389  — 

amj  special  detenmudion  of  S2q  ami  p^,  maf/  he   reduced  io  two  fjeneral 
forms  of  algebraia  poli/nomicd  besides  other  fractional  forms. 
The  first  of  these  forms  is 

in  which  m  is  either  2,  1,  |,  or  |^. 
The  second  form  is 

F^b,  r  "'-3  +  ^3  r'"-3  +  b.  /--s  +  b,  r-'"-3  +  b  r-2— 3^ 

in  wliich  m  is  either  1  or  2.  In  each  of  these  cases  the  term  which 
is  inversely  proportional  to  r  must  be  omitted. 

691.     Iji  the  first  case  of  the  preceding  section,  when  m  is  unity, 
the  equation  (07892)  acquires  the  form 

„=  r M_, . 

It  is  obvious  from  inspection  that  whenever 

a^  =  b^, 

is  positive,  a  portion  of  the  curve  extends  to  infinity;  but  when- 
ever «4  is  negative,  the  curve  is  of  finite  extent.  It  is  also  apparent 
that  whenever 

a^  —  iB'-pl, 

is  positive,  a  portion  of  the  curve  terminates  in  an  infinitely  com- 
pressed coil  about  the  origin,  that  no  portion  of  the  curve  can  ap- 
proach the  origin  except  through  such  a  coil,  and  that  when  a  is 
negative,  the  curve  does  not  pass  through  the  origin. 
If  all  the  roots  of  the  equation 

are  imaginary,  a^  and  a  must  be  positive,  and  the  curve   extends 


—  390  — 

continnoiisly  from  the  origin  to  infinity.  If  the  moduli  of  the  roots 
are  h  and  \^  and  the  arguments  a  and  a-^,  and  if  the  following 
notation  is  adopted 

2  a  Og  —  {li?  -j-  1x\)  cii  a^ 

J^-  =:  [p  —  lif  -)-  4  p  li  sin^  k  « , 
^1  =  {p  —  li^"  -\-^p  K  siii^  h  u^-, 
i>-  =  {fi  —  lif  A^^.qll sin-  h  (i , 
B{z^[q  —  1i^^  -\-iqh^  sin^  a^  «i, 

^li  (?  —  r) 
COSZ=:^^, 

and  if  ^q  is  the  value  of  (3  when  r  vanishes,  the  equation  of  the 
curve  is 

2  pl{p  —  g)- cos' do         ^^  ^  {p—q)cos-do      ^^  "   '^^  "^     ' 


T 


p  tan  (/q 


y/  ( 1  —  sin^  /  sin^  ^,)  ^,  (—  cosec^^  i\ ,  i^ ) 


,        Y^  (1  —  sin-  ^  sin-  ^)  —  y/  (1  —  sin^  i  sin-  ^y) 
°  V/  (sin^  ^  —  sin-^ 


and  the  expression  of  the  time  is 


A  A^  (f  tan  d^  (t  —  t)     I  p'^-\-  <f  cos^  i  iwC-  6q q  {p  cot-  /9o  +  *?)  V  iP'  H~  T  ^^^^  *  ^^^'^'  ^o) 

+ 


,  7  — t)     / 

/((?—;>)  (<?  —  »■)     V         fcot'do^q'      ~  (q  +  r)  (p^  coi' d,-{- q')l 

q{p--\-pq)\J  (p'  +  <?"  cos-  /  tan-  /?„) 


+  lo 


(^  _r)  tan^  dolp"  cot-  ^0  +  9')^ 

\/[(p"  cot-  /9|)  +  ?"')  (1  —  sin- ^■  sin- 19) ]  —  y/  (/r  cot"  6^ -\-  (f  cos- /) 


—  391  — 
The  elliptic  integrals  disappear  when 

which  case  has  already  been  discussed  in  §  686.     They  also  disap- 
pear when  the  imaginary  roots  are  equal,  in  which  case 


Qi         a 

SO  that  if 


^=g=4^,-8v/(«^,); 


i?  .-=  2  (/g  a^  7^  +  al  )'-\-2  a^  a^ , 
the  expressions  for  y  and  t  are 

(p  _  «  ^  ZL  locr  r Mil tanf-l^        (4a,r+a3)v^^i 

'  \J  a      °  li        y/ [a^ag  04  (16  a  a^  —  «i«3)]    '  VC^sClGaa^  —  Oi  o 


When  two  of  the  roots  of  the  equation  (oSQag)  are  real  and 
two  are  imaginary,  if  both  the  real  roots  are  negative,  a^  and  a 
must  be  positive,  and  the  curve  extends  continuously  from  the 
origin  to  infinity.  If  one  of  the  real  roots,  denoted  hy  f'l,  is  posi- 
tive, and  the  other,  denoted  by  rs,  is  negative,  and  if  «4  is  positive, 
the  curve  extends  to  infinity  at  each  extremity,  and  rj  is  its  least 
distance  from  the  origin ;  but  if  a^  is  negative,  the  curve  is  finite, 
terminates  at  each  extremity  in  the  origin,  and  )\  is  its  greatest 
distance  from  the  origin.  If  both  the  real  roots  are  positive  and 
if  «4  is  also  positive,  the  curve  consists  of  two  portions,  one  of 
which  extends  to  infinity  at  each  extremity,  and  the  greater  real 
root  r^  is  its  least  distance  from  the  origin,  while  the  other  portion 
is  finite,  terminates  at  each  extremity  in  the  origin,  and  r^  is  its 
greatest  distance  from  the  origin  ;  but  if  a^  is  negative  the  curve 
consists  of  a  continuous  portion  of  which  rj  is  the  greatest,  and  r^ 
the  least  distance  from  the  origin.     If  h  is  the  modulus  and  a  the 


—  392  — 

argument  of  one  of  the  imaginary  roots,  the  following  notation 
may  be  adopted. 

^^_/ic-as/-i_^tani£  c-/3^-i, 

ra  —  li  C"  ^"^  =  A  cot  As  c'\'^~^ , 
r^  —  /^c~'^^'~^  =  ^cot  if:  c~h^~'^, 
Vi  cot  i£  =  /cot  hy , 
fa  tan  -2-£  =  /tan  2/. 

AVhen  a^  is  positive,  if  <3  and  i  are  determined  by  the  equations 

tan  J  (3  cot  ^  £  =  i  /  (   _  'j , 
or 

r  sin  )» cos  y  —  cos  (9 sin  ^  (^  -|-  J')  ^''i  2  i^  —  7) 

I  sin  5        cos  c  —  cos  (?        sin  ^  (<?  +  c)  sin  ^  (^  —  e) ' 

the  equation  of  the  curve  is 

lAsinein) — tt)\/a.  1 — cosscosy^—  ,  .  \  rsn  /  •  o  '  •}•  w 
iiH, — -^^=z -. ^?F.(3— cotrfcosr— cos£)'2J'i( — snrrsnr/,e) 

2 Pi  siny  I  \        I  /      »\  '    / 

cosj' — cose       rn      rji  sin;'sin/9\/[l — sin^isin^r)  (1 — sin^^sin^^)] 
y/  ( 1  —  sin- z"  sin-  y)  1  -j-  cos  y  cos  6  -\-  sin'-  /'  sin-  y  sin-  d  ' 

and  the  value  of  {t  —  t)  is  derived  from  that  of  {(p  —  a)  by  multi- 
plying by  ^,  and  interchanging  /  and  £.  It  is  apparent  that  £  is 
obtuse  and  exceeds  y,  and  that  upon  the  finite  portion  of  the 
curve  ^  extends  from  zero  to  y,  while  upon  the  infinite  portion, 
it  extends  from  £  to  n. 

When  a^  is  negative,  i£  6  and  i  are  determined  by  the  equations 

tan  i6  cot  ^t=i\/  y  _  y? 


—  393  — 
or 

r  sin  y        1  —  cos  r  cos 


/  sin  e        1  —  cos  £  cos  d ' 

the  equation  of  the  curve  is 

/^  sine  sin  2  7  ((]r  — «)  y/ — a^  .    „      ^         ,     ,  \ 'S(>  t      l       w 

^^1 '-^ =  sm- ;'  ^',  ^  +  (cos ;'  —  cos  e)  ly,  (cot ;',  ^) 

(cos  y  —  cos  g)  cos  7  ^^^^[-1]     /  /cot^  7  -f  sin''  A 
'       y/  (cot'^ ;.'  -]-  sin-  /)      '  \     Vcot-  /!/  -j-  cos'-^i/  ' 

and  the  value  of  (/  —  t)  is  derived  from  that  of  (9  —  a')  by  multi- 

7-2 

plying  by  -r—.,  and  interchanging  /  and  e. 

The  elliptic  integrals  disappear  when  the  two  real  roots  are 
equal.  In  this  case,  ^/^  is  positive,  and  the  curve  is  continuous  from 
the  origin  to  infinity.     With  the  notation 

Pi-z^i^  -\-  /r  —  Irh  cos  «  =  (r  —  li  cos  «)"-!-  /r  sin-  a , 
E\=^]\-\-  li-  —  2  y'l  li  cos  «  =  (;*!  —  /i  cos  u^"  -\-  li"  sin- « , 

the  equation  of  the  curve  is 

ri  (qj  — a)  v'cf4 1  Tn,.[-i]  ^^— ^'^Q^^^         1    Tor.[-i]  /^^  —  ^  cos  «  (r -[- rQ  +  r  ri 

and  the  expression  of  the  time  is  given  by  the  equation 

When  r/4  vanishes,  if  rj  is  the  real  root  of  the  equation  (SSOgg), 
the  curve  consists  of  a  single  portion  which  extends  from  the 
origin  to  infinity  when  i\  is  negative,  in  which  case  ^3  is  positive. 
But  if  i\  and  a^  are  both  positive,  the  portion  extends  to  infinity, 
and  'i\  is  its  least  distance  from  the  origin  ;  if  i\  is  positive  while 
ffg  is  negative,  each  extremity  of  the  curve  terminates  in  the  origin, 
and  Vy  is  its  greatest  distance  from  the  origin. 

50 


—  394  — 


When  cfg  is  jjositive,  if  tl  and  i  are  determined  by  the  equations 
the  equation  of  the  curve  is 

i?((p-a)v/a3_         erf,   A_^\±J^C^^\(ll^]^    ^^ 

B{r^  —  B'^f ,       [_ij  sin^ v/(rf-|-^^  — 2^ViCOs20 

+  ^/^;r(,|qr^53:Ti?^7i  cos  2  O]  ^'^^  2  i? sl{_r,  (l  —  sin^  ^•  siiV^  6^)]      ' 

and  the  expression  for  the  time  is 

(^_t)  v/«3=  (5  +  ^)3^.^  — 2i?^\^  + 2i? tan |^v'(l—sin^-sin\4). 

When  ^3  is  negative,  if  (3  and  i  are  determined  by  the  equations 

tan^  I  (3  =  "^;^~  ? 
the  equation  of  the  curve  is 

Oi+^) 27? _— J,6—      2^,^      J,[^  4i?^r,   '^J 

i? (r,  +  i?-^^  rp^ .  r_ij  sin ^  v^  in  J^B'^2B'r, cos  2  0 

+  ^[r.j(rf+^*+2^ViCos20]  2 ^ y/  E^i  (1  — sin^ 2^" sin^  ^)]      ' 


and  the  expression  for  the  time  is 

(^;_T)^_a3==(^_7?)?J^,;(3  +  2i?^,^— 2^tanidv/(l  — sin2^sin2^). 

When  all  the  roots  of  the  equation  (SSOgg)  are  real,  if,  beginning 
with  the  greatest,  they  are  arranged  in  the  order  of  algebraic 
magnitude,  they  may  be  denoted  by  7\,  r^,  r^,  and  r^.  If  they  are 
all  negative,  the  curve  consists  of  a  single  portion  which  extends 
from  the  origin  to  infinity.     But  if  Vi  is  the  only  positive  root,  the 


curve  consists  of  a  single  branch,  which  extends  by  the  same  law 
as  that  expressed  in  (39I20).  If  r^  and  r^  are  positive,  while  the 
other  two  roots  are  negative,  the  curve  consists  of  one  or  two  por- 
tions, according  to  the  same  principles  which  distinguish  the  forms 
of  (39I25).  If  ^4  is  the  only  negative  root,  and  if  a^  is  positive, 
the  curve  consists  of  two  portions,  one  of  which  extends  to  in- 
finity, and  Tj  is  its  least  distance  from  the  origin,  while  the  other 
portion  is  finite  and  limited  by  the  circumferences  described  about 
the  origin  as  centre,  with  r2  and  r^  as  radii ;  but  if  a^  is  negative, 
one  portion  terminates,  at  each  extremity,  in  the  origin,  and  rs  is 
its  greatest  radius  vector,  while  the  other  portion  is  contained  be- 
tween the  limiting  circumferences  of  which  rj  and  r2  are  the  radii. 
If  all  the  roots  are  positive  and  if  a^  is  also  positive,  the  curve 
consists  of  three  portions,  one  of  which  extends  to  infinity  and  ri 
is  its  least  distance  from  the  origin,  a  second  portion  is  limited  by 
the  circumferences  of  which  r2  and  r^  are  the  radii,  and  the  third 
portion  passes  through  the  origin  at  each  extremity,  and  i\  is  its 
greatest  radius  vector ;  if  a^  is  negative,  the  curve  consists  of  two 
portions,  one  of  which  is  limited  by  the  circumferences  of  which  ?'i 
and  ra  are  the  radii,  and  the  other  by  the  circumferences  of  which 
rs  and  r^  are  the  radii. 

When  «4  is  positive,  the  following  notation  may  be  adopted. 

Ti  —  Tg  =  A  tan  i  £  tan  i  i] , 
t'l  —  r2  =  A  cot  i  e  tan  J  i]i, 
r^  —  r^^  A  tan  h  £  cot  \  ^] , 

rg  —  9\  =  yl  cot  i  c  cot  h  i]i , 


-k  Tt  —  £  ; 


which  give 


tan?/ 
tun  //i ' 


—  396  — 

or  sin  (ij,  —  t]) 

COS  S  =    .  ,      ,- . 

sin  (?/i  +  ,i) 

For  the  portion  of  the  curve,  which  is  contained  between  the  cir- 
cumferences of  which  ra  and  r^  are  the  radii,  the  notation 

ra  sin  i  7^  =  / sin  i  x, 

rg  cos  ^i]=ilco8ix, 

^  ^        tan  i  A/ (n  —  ?•)  ' 

gives 

r sin  ^  (j^i  -f  x)  -f  sin  1  (j^i  —  x)  sin  /9 

/        sin  i  (7/1  +  ,^)^  sin  H'/i  — ^/)  sin^" 

The  equation  of  the  curve  is,  then, 

^(qp  — «)v/a,_sinK>/.— ^/)g,  ^,         siny,s\n  ^  (,j,  —  x)       ^T      sin^^fa  — x)     1 
2pllcosi  sini(,/i_x)      '       I   sini(7/i— x)sini(j?i4-x)     ^      sin^^^j^i+x)'    J 

sin  ^  (?/  —  x)  y/  (sin  y,  cosec  x)  ^_  ^ ^  y'  ( 1  —  cos^"  tan^  d) 

V/[sin //, sin x  —  sin^ i  sin^ i  (,^, -[- x) ]    ''^^^        ^ [1— sin2^•^in2  J-  (7/^+x)  cosec //^  cosec x] ' 

and  the  expression  for  the  time  may  be  obtained  from  this  value 
of  {(p  —  a)  by  interchanging  x  and  1]  and  multiplying  by -f^. 

The  nature  of  the  motion  through  the  space  exterior  to  the 
circumference  of  which  7\  is  radius,  and  within  the  circumference 
of  which  r^  is  radius,  may  be  derived  from  equations  (3965_i5)  by 
changing  r^  to  r^  and  r^  to  9-4^  and  augmenting  each  of  the  angles 
t]  and  X  by  the  magnitude  tt. 

When  «4  is  negative,  the  following  notation  may  be  adopted, 

ri  —  rg  =:  A  tan  -|  e  tan  i  1], 
H  —  ^4  =  ^  t^n  h  £  cot  g  1; , 
ri  —  i\=z  A  cot  2-  €  tan  ^  i^i , 
^2  —  ^3  =  -4  cot  I  £  cot  1 1^1, 
^  ==  ^  TT  —  I  c . 


The  nature  of  the  motion  between  the  circumferences  of  which 


—  397  — 

r^  and  ?-^  are  the  radii  may,  then,  be  expressed  by  the  equations 
(3965_i5),  provided  that  rg  is  changed  to  r^,  and  ?-2  to  r^  and  the 
sign  of  a^  is  reversed.  The  character  of  the  motion  between  the 
circumferences  of  which  r^  and  r^  are  the  radii,  may  be  expressed 
by  the  same  equations  with  the  change  of  Vi  to  rg,  and  of  rg  to  ?\, 
the  reversal  of  the  sign  of  a^,  and  the  increase  of  each  of  the 
angles  1]  and  x  by  Ji. 

The  elliptic  integrals  disappear  when  two  of  the  roots  are 
equal ;  in  this  case,  if  Ti  denotes  one  of  the  equal  roots,  and  if  11^  is 
the  quotient  of  the  division  of  the  first  number  of  (08929)  by 
(r — rif  so  that  the  form  of  E^  is 

B^  =  /?2  r^  -{-  hi  r  -\-  h, 

the  notation  may  be  adopted 

Bl  =  /?2  ri  -j-  hi  Vi  -\-  h, 
2  h  -\- hiT  =  2  B  sj—  h  t3in{&  sJ—h)  =  —  2B  ^ h  Tan{6>^h), 
hi  +  2  7^2  r  =  2 i? s/—ho^  tan  (L\iJ—h,)  =  —  2  ^  yZ/^a  Tan  (^2 >Jh), 

^_in±^^±Ml=  _  v/-  1  tan  (Bi^^-1)=  Tan  (7?^ ^,) ; 
the  equation  of  the  curve  is 

and  the  expression  of  the  time  is 

t  —  T=z6r^-\-ri^i. 

When  «4  vanishes,  if  a^  is  positive,  the  notation  may  be  adopted 

ri  —  r^  =  B^  tan^  j  e , 

Ti  —  rsz=  B^  cot^  i  fc- , 
r^  cos^  it  =  I cos^  I X , 
rg  sin^  h=:l  sin^  ^  x , 

e  z=:  2'  TT  —  £  ; 


—  398  — 

and  for  the  portion  of  the  curve  contained  between    the    cu^cum- 
ferences  of  which  rg  and  y-g  are  the  radii, 


tan^  (}n  —  i^)  =  tan^  it  ''-^. 
The  equation  of  this  portion  of  the  curve  is,  then, 

(^ -^, ^ cj>se  /^  _  COS^X  g,  ^^^, 

2  pi  I  COS  ^  COS  X  '     \  cos  yJ         ^  ^     ' 

I  cosx  —  cose  ,      [_ij     //    1 -|- cos^ t  tan^ ^    \ 

■^  sin  X  y/  (cos"  t  —  cos^  x)    "^  y    Vcos'  i  —  siir  ^  cot-  x/  ' 

and    the    expression   of  the    time    is   obtained  from  this  value   of 
(f/) — «)  by  interchanging  e  and  x  and  multiplying  by  ^\ 

Upon  the  portion  of  the  curve  exterior  to  the  circumference, 
of  which  Vx  is  radius,  the  notation 

r  —  ri  =  B"  tan^  ( i  tt  —  ^(1)  =  ^f^^  B\ 


gives  for  the  equation  of  the  curve 

{cp  —  a)sla,^    gi.^ 2  B^ 

2  pi  cos  i  r^  —  B'        (ri  —  B-^){r^  +  i?^) 


^^[-c-s^:)^'] 


B 


+ o  in [- 1]  ^  ^  V/  ('"i  +  ''i  cos'  /  tan'  Cjp) 


"    sJlr,{\r^B^co%''i  —  {i\  —  BY&m-'i)-]  >J  [^ir^  BHo^H  — {r^—By s\n^ iV 

and  for  the  expression  of  the  time 

(^_^)sec^  =  (ri4-52)3^,.<3  — ^'g,^  +  ^|^V(l  +  cos=^^'tan2^) 

+  2  Z?2  Tant-iJ  y/  ( 1  +  cos^  ^  tan^  d ) . 

If  ^^3  is  negative,  the  notation  may  be  adopted 

ri  —  rg  =  B'^  tan^  ^g , 
'>\  —  ^3  =  -^^  cot^  2«  ; 


—  399  — 

whrch,  combined  with  that  obtained  from  (39727_3i)  by  changing 
i's  into  ;-2  and  rg  into  ri,  gives  (398;)  for  the  equation  of  the  por- 
tion of  the  curve  contained  between  the  circumferences  of  which 
Ti  and  ^2  are  the  radii,  while  the  expression  of  the  time  is  derived 
by  the  process  of  (398i3).  But  with  the  notation  obtained  from 
(398i6)  by  changing  rj  into  rg  and  reversing  the  sign  of  7>-,  the 
equations  (398i9)  and  (39825)  become  the  equation  of  the  curve 
and  the  expression  of  the  time,  upon  the  portion  which  is  con- 
tained within  the  circumference  of  which  r^  is  the  radius. 

The  form  of  the  central  force  which  corresponds   to   the  dis- 
cussion of  this  section  is 

692.     If  7)1  is  2  in  the  first   class  of  §  G90,  the  expression  of 
the  central  force  is 

and  the  forms  of  the  equation  of  the  curve  are  obtained  from 
those  of  §691  by  changing  r  into  r,  and  (cp  —  a)  into  2{(p  —  a). 
But  the  expressions  of  the  time  require,  moreover,  the  substitution 
for  (390.6)  of 

p  —  q     ^ 

for  (39I14)  of 

t-r  =  J  (^-^ ^  tan^-^^  7^r?-^^^±"^-^n  > 

for  (392.3)  of 

for  (393y)  of 

(/  — r)v^  — «,=  ^eF.(3, 


—  400  — 

for  (39823)  of 

(?  — TjVf/i  —  ^ian  ^^^  , 

for  (394i2)  of 

for  (39425)  of 

for  (396i8)  with  the  form  of  (39G23)  of 

{t  —  t)  y/«i=  i  cosz  3^i<3, 
for  (3975)  of 

for  (397.4)  of 

t  —  T=^  -h  6, 

for  (398n)  and  (39825)  of 

{t  —  t)  «3  =  i  cos  i  9=,  ^ , 
and  for  (3992_c)  of 

{t  —  t)  y/  —  ^(^3  =  2  cos  i^\6. 

693.     In  the  special  case  of  §  692,  in  which  F  is  reduced  to  its 
first  term,  so  that 

two  of  the  roots  of  (38929)  are  real  and  two  are  imaginary,  so  that 
the  only  portion  of  §  691,  which  is  applicable  to  this  case,  is  from 
(39I15)  to  (39823).  In  this  case,  moreover,  one  of  the  real  roots  is 
positive  and  the  other  is  negative  if  b^^  is  positive,  so  that  the  curve 
extends  to  infinity ;  but  if  b^  is  negative,  both  of  the  real  roots  must 
be  positive,  so  that  the  circumferences  which  correspond  to  these 
roots  are  the  limits  of  the  curve,  and  S2q  is  negative  and  satisfies  the 
condition 

-i2o>|-^f^(-2J,). 


—  401  — 

694.     In    the    special  case  of  §  692,  in  -which  F  is  reduced  to 
its  second  term,  so  that 

the  equation  (88939)  lias  no  imaginary  roots   of  i^  when 

4o^ 


b. 


>'^1p\ 


When  h^.  i^  positive,  there  is  only  one  real  root,  so  that  the 
curve  extends  to  infinity  from  the  circumference,  which  is  defined 
by  this  root.  When  h^,  is  negative,  all  the  roots  must  be  real,  and 
the  two  roots,  which  are  positive,  define  the  circumferences  which 
limit  the  extent  of  the  curve. 

695.  If  m  is  f  in  the  first  class  of  §  690,  the  expression  of 
the  central  force  is 


F=^  h^  r  ^  -|-  h.i  r  ^  -f-  h^  i'~'^  -)-  ^ 


—3 


and  the  forms  of  the  equation  of  the  curve  are  obtained  from 
those  of  §691  by  changing  /•  into  z-^,  and  9)  —  u  into  §(9)  —  a). 
But  the  formula?  for  the  time  are  more  complicated,  although 
they  are  still  reducible  to  elliptic  integrals.     If,  indeed, 


the  expression  for  the  time  assumes  the  form 

,_    _   C %z^ 

696.     In  the  special  case  of  §  695,  in  which  F  is  reduced  to 
its  first  term,  so  that 

F^hr-^, 

the  conditions  of  the  form  of  the  curve  are  the  same  with  those 

51 


—  402  — 

expressed   in    (40023_3o),   but   instead  of  (400n,),   the  limitation  of 
Hq  when  h^  is  negative  is 

697.  In  the  special  case  of  §  695,  in  which  F  is  reduced  to 
its  second  term,  so  that 

the  equation  (08929)  bas  no  imaginary  roots  of  y'  r^,  when 

In  the  special  case,  in  which  F  is  reduced  to  its  third  term,  so  that 

F=h,r-i, 
the  equation  (08929)  l^^^s  no  imaginary  roots,  when 

In  each  of  these  cases,  when  £1^  is  negative,  there  is  only 
one  real  positive  root,  so  that  the  curve  extends  to  infinity  from 
the  circumference  which  is  defined  by  this  root.  When  ii^  is 
positive  all  the  roots  must  be  real,  and  the  two  roots,  which  are 
positive,  define  the  circumferences,  which  limit  the  extent  of  the 
curve. 

698.  If  m  is  i  in  the  first  class  of  §  690,  the  exj)ression  of  the 
central  force  is 

and  the  forms  of  the  equation  of  the  curve  are  obtained  from  those 
of  §  691  by  changing  r  into  y/;-  and  (^  —  a)  into  5  (9)  —  a).     But  if 


—  403  — 
the  expression  of  the  time  assumes  the  form 


^  —  "^=[77- 


^^a,~J-^a,z'  +  "i^  +  ^) 


699.  In  the  special  case  of  §  698,  in  which  F  is  reduced  to 
its  first  term,  so  tliat 

and  in  that,  in  which  it  is  reduced  to  its  third  term,  so  that 

two  of  the  roots  of  (3892o)  are  real  for  \J  r,  and  two  are  imaginary, 
so  that  the  only  portion  of  §  691,  which  is  applicable  to  this  case, 
is  from  (39I15)  to  (39024).  ^^^  this  case,  moreover,  one  of  the  real 
roots  is  positive  and  the  other  is  negative  if  S2q  is  negative,  so  that 
the  curve  extends  to  infinity  ;  but  if  S2q  is  positive,  both  of  the  real 
roots  must  be  positive,  so  that  the  circumferences,  which  correspond 
to  these  roots,  are  the  limits  of  the  curve,  and  in  the  former  of 
these  cases  ^3  is  negative  and 

while  in  the  latter  case  hi  is  negative  and 

700.  In  the  second  class  of  §  690,  when  m  is  unity,  the  equa- 
tion (37822)  of  the  curve  assumes  the  form 

SO  that  it  can  always  be  obtained  from  the  expressions  of  (/  —  r) 
in  §  692,  by  multiplying  either  of  those  expressions  by  4^>^  When, 
in  this  class,  the  curve  terminates  in  the  origin,  it  does  not  usually 


—  404  — 

pass  through  the    condensed  coil    of  §  691.     The    formula  for   the 
time  is 

;_^=  f -= 

The  form  of  the  force,  which  corresponds  to  this  case,  is 


F=^h  r-^+  h  r-^  +  h  y-'+h  r 


5 


701.     In  the  special  case  of  §  700,  in  which  F  is  reduced  to  its 


third  term,  so  that 


F=^' 


4? 


one  of  the  roots  of  (08929)  is  zero,  and  the  condition  that  all  the 
roots  are  real  is 

i_^^ 

When  /2q  is  negative,  if  h^  is  positive,  the  curve  extends  to 
infinity,  in  the  space  exterior  to  the  circumference  of  which  the 
positive  root  of  (08929)  is  the  radius;  hut  if  bx  is  negative,  the 
curve  extends  from  the  origin  to  infinity,  if  two  of  the  roots  are 
imaginary,  but  if  all  the  roots  are  real,  one  portion  is  exterior  to 
the  circumference  of  which  the  greater  positive  root  is  radius  and 
extends  to  infinity,  while  the  other  portion  is  contained  within 
the  circumference  of  which  the  smaller  positive  root  is  the  radius, 
and  this  portion  passes  through  the  origin.  When  Hq  is  positive,  hi 
is  negative,  and  the  curve  passes  through  the  origin,  and  is  con- 
tained within  the  circumference  of  which  the  positive  root  of 
(38929)  is  the  radius.  This  case  of  force  has  been  analyzed  by 
Stader. 

702.     In  the  special  case  of  §  700,  in  which  F  is  reduced  to 
its  last  term,  so  that 


—  405  — 

all  the  roots  of  (oSOgg)  are  imaginary  when  S2q  and  b  are  both 
positive.  When  /2,)  is  positive,  therefore,  b  must  be  negative  and 
the  curve  is  contained  within  the  circumference  of  which  the  pos- 
itive root  of  (oSQog)  is  the  radius.  When  S2q  is  negative,  if  b  is 
positive  the  curve  extends  to  infinity  in  the  space  exterior  to  the 
circumference  of  which  the  positive  root  is  radius;  but  if  b  is 
negative,  the  curve  consists  of  two  portions,  one  of  which  extends 
to  infinity  in  the  space  exterior  to  the  circumference  of  which  the 
greater  real  root  is  radius,  while  the  other  portion  passes  through 
the  origin  and  is  contained  within  the  circumference  of  which  the 
smaller  root  is  radius ;  or  it  extends  from  the  orio;in  to  infinitv. 

703.     When  m  is  2  in  the  second  class  of  §  600,  the  form  of 
the  force  is 

F=b,r-]-b.,?--^-\-b,i'-''-}-br-', 

and  the  equation  of  the  curve  can  be  obtained  in  each  case  from 

that  of  §092,  hy  niultipl^dng  [t  —  t)  by  2y/|,  and  changing  t  —  t 

into  (f>  —  a,  and  ;•  into  r^. 

If 

.  =  ;^ 

the  formula  for  the  time  is 


^~l\/\'a,z*  +  a,:^' 


V/  [^'4  *"*  +  03-''  +  a.,  02  +  «i  -  +  «] 

704.     In  the  special  case  of 


1-^ 


there  are  two  imaginary  roots  of  r  when 

h^    G-tyr 

When  wQq  is  negative,  if  b  is  positive  the    curve    extends   to 


—  406  — 

infinity  in  the  space  exterior  to  the  circumference  of  which  the 
real  root  of  (oSOso)  is  the  radius;  but  if  h  is  negative,  and  if  all 
the  roots  of  (oSQgg)  are  also  real  and  two  of  them  positive,  the 
curve  consists  of  two  portions,  one  of  which  extends  to  infinity  in 
the  space  exterior  to  the  circumference  of  which  the  greater  posi- 
tive root  is  radius,  while  the  other  portion  passes  through  the 
orio-in  and  is  contained  within  the  circumference  of  which  the 
smaller  positive  root  is  radius  ;  but  if  neither  of  the  roots  is  positive 
when  ^  and  il^  are  both  negative,  the  curve  consists  of  a  single 
portion  w^hich  extends  from  the  origin  to  infinity.  When  il^^  is 
positive,  h  must  be  negative  and  the  curve  consists  of  a  single 
j)ortion  which  passes  through  the  origin  and  is  contained  within 
the  circumference  of  which  the  positive  root  is  radius.  This  law 
of  force  has  been  analyzed  by  Stader. 

705.  Another  class  of  central  force,  in  which  the  integration 
can  be  performed  by  elliptic  integrals,  corresponds  to  the  form  of 
the  potential 

^.  __  h^  r'' '"  4-  h.  1^ '"  -\-  b.j  r- '"  -\-  b^  r'"  -f-  b 

in  which  m  may  be  either  1  or  2.     If,  in  these  forms 

y  ^,w 

^  —  /     , 

Z''"  =  a^  z'^  -\-  cIq  z^  -\-  6(2  z"  -^a^z^  a 

=  {2.b^ 7-'"'  +  2  ^3 r-^'"  +  2 b. r '«  +  2  ^^ r -f  2 h) 
-[2.n,^^  +  ^p\){r-+hf, 

the  equation  of  the  curve  assumes  the  form 
and  the  expression  of  the  time  is 


m 


—  407  — 

706.  The  following  graphic  construction  gives  an  easy  geo- 
metrical process  for  tracing  the  various  cases  of  limitation  of  the 
extent  of  the  path  described  under  the  action  of  a  central  force, 
and  especially  for  finding  by  inspection  the  effect  of  the  values 
of  S2q  and  j}^  upon  the  limits  of  the  curve.     If 

1 

v 

construct  the  curve  of  which  the  equation  is 

which  may  be  called  iJie  ^potential  curve,  draw  the  straight  line  of 
which  the  equation  is 

and  the  points  of  intersection  of  the  straight  line  with  the  potential 
curve  give  the  values  of  x  for  the  limits  of  the  path  of  the  body. 
The  path  corresponds  to  those  portions  of  the  potential  curve 
which  lie  upon  that  side  of  the  straight  line,  which  is  positive  with 
respect  to  the  direction  of  the  axis  of  y. 

707.  A  term  of  il  may  be  omitted  in  the  preceding  construc- 
tion which  is  inversely  proportional  to  the  square  of  the  radius 
vector,  and  its  negative  may  be  combined  with  that  term  of  the 
equation  of  the  straight  line  which  determines  its  direction.  The 
omitted  term  corresponds  to  a  term  of  the  force  Avhich  is  inversely 
proportional  to  the  cube  of  the  radius  vector,  and  which  m.ay  be 
represented  by  (08017)  ;  and  the  corresponding  equation  of  the 
straight  line  is 

708.  It  is  evident  from  the  preceding  construction,  that  if  the 
poteiiiial  curve  has  no  point  of  contrary  flexure,  and  if  its  convexity  is  turned 


408  — 


in  the  direction  of  the  positive  axis  of  y,  the  path  of  the  lochj  can  only 
consist  of  a  single  portion  tvhich  may  have  either  an  outer  or  an  inner  limit, 
or  it  may  have  neither  or  both.     This  case  includes  all  forces  of  the  form 


F=hr-  +  l,, 


r 


in  which  h^  and  m  -\-  3  have  the  same  sign. 

Bid  if  the  potential  curve  lias  no  point  of  contrary  flexure,  and  if-  its 
convexity  is  turned  in  the  direction  of  tJie  negative  axis  of  y,  t/ic  path  of 
tlie  body  may  consist  of  a  single  portion  ivldcli  has  either  an  outer  or  an 
inner  liiuit,  or  it  may  liave  neitlier,  or  it  may  consist  of  two  separate  por- 
tions of  ivldcli  one  lias  only  an  outer  and  the  otlier  only  an  inner  limit.  This 
case  includes  all  forces  of  the  form  (4O85),  in  which  ^1  and  niA^  3 
have  different  signs. 

709.  Those  portions  of  the  potential  curve,  in  Avhich  y  and  x 
simultaneously  increase,  correspond  to  the  distances  from  the  centre 
of  action,  at  which  the  force  is  attractive,  so  that  the  convexity  of 
the  path  of  the  body  is  turned  away  from  the  origin.  The  portions 
of  the  potential  curve,  in  which  y  decreases  with  the  increase  of  x, 
correspond  to  the  distances  from  the  centre  of  action,  at  which  the 
force  is  repulsive,  so  that  the  convexity  of  the  path  of  the  body  is 
turned  towards  the  origin.  Any  point,  therefore,  at  which  the 
potential  curve  is  parallel  to  the  axis  of  x,  and  the  ordinate  is  either 
a  maximum  or  a  minimum,  corresponds  to  a  distance  from  the 
origin,  at  which  the  central  force  changes  from  attraction  to  repul- 
sion, and  the  path  of  the  body  has  a  point  of  contrary  flexure. 

710.  If  for  an  infinitesimal  value  of  r  denoted  by  ?",  il  assumes 
the  form 

12  =  /?:  2% 

the  path  of  the  body  cannot  pass  through  the  origin  if  n-\-1  is 
positive  or  if  k  is  negative,  except  in  the  former  case,  when  p^  van- 


—  409  — 

ishes  and  n  is  positive  while  I2q  is  negative,  or  ?^  is  negative  while 
k  is  positive;  but  if  k  is  positive  and  ii-\-2  negative,  the  external 
portion  of  the  path  passes  through  the  origin,  and  after  passing 
through  the  origin,  the  continuity  of  curvature  is  destroyed  and  the 
path  becomes  a  straight  line. 

711.  If  for  an  infinite  value  of  r,  denoted  by  the  reciprocal 
of/,  II  assumes  the  form  (4O828),  the  path  of  the  body  cannot  extend 
to  infinity  when  n  and  Ic  are  both  negative,  or  when  n  and  S2q  are 
both  positive,  or  when  n  vanishes  and 

but  the  external  portion  of  the  path  extends  to  infinity  when  n  is 
negative  and  k  positive,  or  when  «  is  positive  and  S2q  negative,  or 
when  n  vanishes  and 

712.  If  a  line  is  drawn  parallel  to  the  axis  of  x  at  the  dis- 
tance S2q  from  this  axis,  and  assumed  as  a  new  axis  of  a^i,  and  if 
^1  and  ^2  ^i"®  the  corresponding  ordinates,  respectively,  of  the 
straight  line  (407i4)  ^^d  of  the  potential  curve,  the  value  of  the 
angle,  which  the  path  of  the  body  makes  with  the  radius  vector, 
is  given  by  the  equation 

V  y.        B 

which  admits  of  simple  geometrical  construction.  If  z-^  denotes  the 
subtangent  of  the  potential  curve  upon  the  axis  of  x^,  the  projection 
of  the  radius  of  curvature  of  the  path  of  the  body  upon  its  radius 
vector  is 


which  is  constructed  without  difficulty.  By  the  combination  of 
these  two  constructions,  the  path  of  the  body  may  be  obtained  with 
sufficient  exactness  for  most  purposes  of  general  discussion. 

52 


—  410  — 

713.  When  the  origin  is  infinitely  remote  from  the  body, 
the  forces  o/"  §  676  are  jyarallcl,  and  the  plane  of  motion  is  'parallel  to  the 
direction  of  action,  and  the  equation  (0785)  gives,  if  the  axis  of  z  is 
supposed  to  have  the  same  direction  with  the  force, 

—  2cot;  A,% 


of  which  the  integral  is 

in  which  a  is  an  arbitrary  constant,  which  is  always  positive,  and 
this  is  the  equation  of  the  path  of  the  lody  referred  to  the  same  coordi- 
nates with  those  of  §  571. 

714.     In   the  case  of  a  constant  force,   the   preceding   equation 
assumes  the  forms 


,  .  a 


a 
2a 


^         g  sin  ~ 

so  that,  in  this  case,  the  path  is  a  parabola. 

715.     The  velocity  in  the  direction  of  the  axis  of  x  is 

2^  sin  ,^  :=  sin  I  V  ( 2  ^2  —  2 12 0 )  ==  v^  ( 2  a ) , 

so  that  this  velocity  is  constant,  and 

The  equation  of  the  curve,  expressed  in   rectangular   coordi- 
nates, is 


—  411  — 

716.  If  a  potential  curve  is  constructed  by  the  equation 
(407io),in  which  y  maybe  changed  into  x,  and  il  retained  as  a 
function  of  ^,  the  limits  of  the  path  of  the  body  are  defined  by 
the  intersection  of  the  potential  curve  with  a  line  drawn  parallel 
to  the  axis  of  z  at  the  distance  (/2q  -\-  ci)  from  this  axis.  The  por- 
tions of  the  potential  curve  which  correspond  to  the  path,  lie  in 
a  positive  direction  from  the  intersecting  line. 

717.  If  the  force  of  §  713  has  the  form 

the  equation  of  the  path  is 

^,  ^  +  ^  =  v/  (^^  +  2  ^1  "'4  +  2  \  a)  sin  [{x,-x)  y/ (-  A.)] 
=  v/(^^  +  2Z.,/2o+2J,«)Cos[g^-2-o)y/A] 

718.  If  the  force  of  §  713  has  such  a  form  that 

^^^  —  {z+hy 

the  notation 

h  =  {k'  +  J^^){n,  +  a), 
gives,  for  the  equation  of  the  path, 

which  is  easily  transformed  into  the  forms,  which  are  appropriate 
when  the  radicals  become  imaginary. 

719.  In  the  case  of  a  surface  of  revolution^  and  a  force  which  is 


—  412  — 

directed  to  a  point  upon  the  axis  of  revolution,  the  notation  of  §  576 
gives 

—  =  !i  sin  1  =  iL^  D, ", 

A=^uv  sin  ,^  =  ir'  Dt  ", 

so  that  the  elementary  area  described  hj  the  projection  of  the  radius 
vector  upon  the  2)lc('^ie  of  i^y  is  constant. 

720.  The  notation  of  §  578  gives 

^  V  2u^  (Si  —  Ho)  —  A^        ' 

^  ^        M  V  2  ?/  (.Q  —  .Qo)  —  ^2 > 
•and,  in  the  case  of  parallel  forces 

■^-^  —  -^-^\2u^{9.  —  Slo)—A^' 

jy  u_ -^  Ag 

^  ^  ~  w  y/  [2  M=^  (.Q  —  ^4)  —  ^']  * 

721.  Upon  the  surface  of  revolution  which  is  defined  by  the 
equation 

the  path  of  the  hody  maJces  a  constant  angle  ivith  the  meridian  curve.  In 
the  case  of 

B^A, 

the  path  is  perpendicular  to  the  meridian,  and  is  a  circle  of  ivhich  the 
plane  is  horizontal. 

Whatever  is  the  value  of  B,  for  the  point  at  which  v  vanishes  u 
is  infinite,  while  v  is  infinite  when  u  vanishes. 

Ujmn  any  other  surface  of  revolution  about  the  same  axis,  the  in- 
clination  of  the  path   of  the  body  to   the   meridian  arc  is  the   same  ivith 


—  413  — 

the  corresponding  inclination  upon  the  surface  of  equation  (4122i)  at  the 
common  circle  of  intersection  of  these  two  surfaces.  Hence  the  limits  of 
the  path  upon  the  given  surface  of  revolution  are  its  intersections  ivith  the 
surface  of  equation 

iiv  =  A, 

and  the  path  extends  over  that  portion  of  the  given  surface^  ivhich  is  ex- 
terior to  this  surface  hj  ivhich  the  limits  are  defined. 

722.  In  the  case  of  a  heavy  body  the  equation  (4122i)  be- 
comes 

u^z  =  —  . 

723.  In  the  case  of  a  heavy  hodg  upon  a  vertical  right  cone,  if 
the  hodg  moves  upon  the  inverted  p>art  of  the  cone,  the  path  has  an 
upper  and  a  loiver  limit ;  hut  if  it  moves  upon  the  part,  v)hich  is  helow 
the  vertex,  the  path  has  an  uptper  limit  from  which  it  extends  doumivards 
to  infinity.  In  this  case,  if  the  notation  of  (341i3)  and  (341i6)  is 
adopted,  if  two  of  the  roots  of  the  equation 

r-  (r  —  ro)  = 


25rsin^acoscc' 

are  imaginary,  which  corresj)onds  to 

^r  sin'' a  cos  « 

if  h  is  the  modulus  and  /:>  the  argument  of  one  of  the  imaginary 
roots,  and  if  7\  is  the  real  root,  the  notation 

r^  —  hc':i-'-^=zB''c''''^-\ 

r^  —  hc-i^^-^  =  B''c-'-'^-\ 

r  —  rj  =  B^  tan^  h  (p , 


—  414  — 
gives  for  the  equation  of  the  path  upon  the  developed  cone 


2B 


^(2rj^—2B'-cos2i)  ^  {■lr^  —  2  B' co&2i) 

and    the   position   of  the   body  at   any  instant   is  defined  by  the 
equation 

-\-2B^  tan  i  9  y/  ( 1  —  sin^ i  sin^  9) . 

If  all   the   roots  of  (4132o)  are  real,  and  denoted  in  the  order  of 
decreasing  magnitude  by  ri,  r.2,  and  r^,  and  if 

ri  —  r^  =  B^tnn^tj, 
r,  — r,  =  B'' cot'' [i, 

r  —  ri=:/':)Han^(T  TT  —  |f/)), 

the  equation  of  the  path  upon  the  lower   portion   of  the  devel- 
oped cone  is 


1"  |-,.^  {r^J^By-co^H  —  4.rlB'-]  ^'^^        \'  (^r,-\-By  cos' i— 4.  r^B'^ 


and   the    position   of  the    body   at   any  instant  is  defined   by  the 
equation 

{t  —  t)  cos  i  y/  (2^  COS  a)  ==  (^  +  ^)  cos^z  ^^,(p  —  2B^,(p 
-\-  \j^-\-  B  sm  (p)  ^  {1  -{-  cos^  i  tan^  (f). 


—  415  — 

The  equation  of  the  path  of  the  body  upon  the  upper  portion 
of  the  cone  is  determined  by  the  combination  of  the  equations 
(414i3_i6)  with 

ri  — r  — i?2 


sni  (p  sni  I 


r,-r^B-^ 


•2  ? 


sin^  aB\J  {g  cos  a) 


:? +oTi[-i]     /  (1+  cos-^  i tan-  y )  [0\-\-B-f  cos-  ^—4 /'i  i?-] 


tanL 


V 


s/[r^(ry-\-B^ycosH—4:7iB^]  V  4ri^2  j 

and   the   position   of  the    body  at   any  instant  is  defined  by  the 
equation 

{t  —  t)  cos/y/(2^cosc«)  =^  —  Cj-\-B)  Qo^iTf^w 

ior)$?  07^-'  /I  —  sin  i  sin  op 

A-  1  B  e,  o)  —  z  B  sm  I  cos  cp  \  /  — ^ — : — ^-^-^ . 

'  '  '    y    \  -\-  sin  I  sin  qp 

The   path  of  the  body  upon  the  upper  portion  of  the  cone 
may  be  exj)ressed  in  a  somewhat  more  simple  form  by  the  equations 


r  =  ?'2  sin^  (f  -\-  rg  cos^  (p , 

and  the  corresjDonding  formula  for  the  position  of  the  body  at  any 
instant  is 

v/(2^cos«)(i'-T)  =  ^§^-2v'(ri-r3)f<9. 

In  the  special  case,  in  which  the  roots  i'2  and  rg  are  equal, 
the  path  upon  the  upper  portion  is  a  horizontal  circle,  and  the 
equation  of  the  path  upon  the  lower  portion  is 

J(d-(?„)  =  tan<-Y'(-i-;,)-\/3tan>-'>y'(-l-?,0, 


—  416  — 

while  the  position   of  the  point  at  any  instant  is  defined  by  the 
equation 

V/(2^cos«)(^-^)  =  2v/(r  +  iro)  +  |v^(-ro)tant-Y'(--^-r}- 

724.  In  the  case  of  a  heavy  hod//  upon  the  surface  of  a  vertical 
paraholold  of  revolution,  of  ivhich  the  axis  is  directed  doivmvards,  the 
path  has  an  upper  limit,  from  ivhich  it  proceeds  downwards  to  infinity. 
If  (33625)  is  the  equation  of  the  paraboloid,  and  if  z^  and  — q  are 
the  roots  of  the  equation. 

%pfjz{z  —  z^)=^-, 

the  path  of  the  body  when 

is  defined  by  the  equations 

^  — ^i  =  (%+^)  tan^c;, 

and  the  position  of  the  body  at  any  instant  is  given  by  the  equation 

I  cos  i{t — T )  d^TT  =  cos^ /  3=,  (f  —  ^ j  9)  -|-  y/  ( cos^ ^  tan^ 9  -|-  sin^ i  sin^ ip ) . 

But  when 

p>q, 

the  path  is  defined  by  the  equations 

^  — 01— (,?i+;9)cot2  9), 
p  —  q  —  {2^-\-p)^\i\^i, 


—  417  — 
and  the  petition  of  the  body  at  any  instant  is  given  by  the  equation 

In  the  especial  case  of 

the  path  of  the  body  is  the  parabola,  which  is  formed  by  the  inter- 
section of  the  paraboloid  with  the  vertical  plane,  of  which  the 
equation  is 

?/cos^=v/(4/  +  ?/^), 

and  the  position  of  the  body  at  any  instant  is  defined  by  the 
equation 

{t  —  r)sj{pg)_^      u 

725.  Ill  the  case  of  a  hcavij  hod;j  vpon  the  mrface  of  a  vertical 
paraboloid,  of  which  the  axis  is  directed  uinvard,  the  path  has  an  upper 
and  a  lower  lindt.  If  y;  is  negative,  (ooGos)  is  the  equation  of  this 
paraboloid,  and  if  — z-^  and  — x^  are  the  roots  of  the  equation 
(416ii),  they  correspond  to  the  limits  of  the  path.  The  path  of 
the  body  is  defined  by  the  formuLi) 

5-  =  —  ^'i  cos^  (/)  —  .?2  sin^  i^) , 
(•?2  —  p)  «iii2  i=-Z2  —  ^1  ? 

and  the  time  is  given  by  the  equation 


418 


THE     SrHERICAL     PENDULUM. 


726.  When  the  surflice  upon  which  the  body  moves  is  that 
of  a  sphere,  the  problem  becomes  that  of  the  sphencal  pendulum.  In 
this  case,  the  path  has  an  upper  and  a  lower  limit.  If  the  centre 
of  the  sphere  is  the  origin,  if  it  is  the  radius  of  the  sphere,  the 
limits  of  the  path  correspond  to  the  roots  of  the  equation 

If  the  roots  of  this  equation  are  %,  2-2,  and  — p,  and  if  the  nota- 
tion is  adopted 

z=.z^  cos^  (p  -\-  ^2  sin^  (f , 
(p  +  ^i)  sin^/^-e'i — *2, 

the  path  is  defined  bj  the  formula 

«v/[2^a-+.,)]  -  ^  5i>  (1=1;,,,)  +  ^  *  (|=iV,), 
and  the  time  by  the  formula 

727.  From  the  equation  (418io),  it  is  easily  inferred  that 

^1^2  + J^''  =  P  {^1  +  ^2), 

that  the  sum  of  %  and  ^2  is  always  positive,  and  that  p  exceeds  Ji. 

728.  It  is  apparent  from  the  inspection  of  (41 821)  that,  if  the 
mutual  ratios  of  ^1  and  the  roots  of  (418io)  are  unchanged,    the 


—  419  — 

time  of  oscilMion  of  the  imididum   is  proportional  to  the  square  root  of 
its  length. 

729.  If  the  length  of  the  pendulum  and  the  sum  of  z^  and  p 
are  given,  it  is  evident  from  (4I821)  that  the  time  of  oscillation 
increases  with  the  increase  of  i^  and  is  a  minimum  when  i  van- 
ishes, that  is,  when 


in  which  case  the   path  of  the    pendulum   is   a   horizontal   circle. 
The  time  of  oscillation  in  this  case  is 

rp '2nR 


Sj[2g{p^z,)j 


The  mutual  relation  of  ^;  and  Zi,  which  is  here  given  by  the  equa- 
tion (4I826),  is 

2p^z-,  +  -, 

n 

whence 

This  value  is  a  minimum,  when 

0j  ^  .3  =  R, 
in  which  case 

which  is,  therefore,  the  greatest  time  of  vibration  ivhen  the  path  of 
the  j)endidum  is  a  horizontal  circle. 

It  is  easy  to  see  that  i  cannot  vanish  for  all  values  of  the 
sum  of ;:»  and  Zi,  but  that  its  least  value  is  determined  by  the 
equation 

sm-  2  ?  :=  4  —  - — -T—^,, 
ip  +  ^i) 


—  420  — 
whenever 

It  is  also  evident  that  the  least  value  of  the  sum  of  p  and  z\ 
which  corresponds  to  any  assumed  value  of  ^  is  given  by  (4]9ao), 
so  that  for  awj  value  of  i ,  ilie  greatest  time  of  vibration  is 

ivMch  increases  tviih  i,  and  is  infinite  lohen  i  becomes  a  rigid  angle. 

When  i  is  an  octant,  the  value  oi  p -\- z^  in  (4193o)  is  a  maxi- 
mum, and  the  corresponding  values  o^  p  -\-  Zi  and  T  are 

p  -\-z^  =  2R 

730.  In  the  discussion  of  the  form  of  the  path  of  the  pendu- 
lum, it  is  convenient  to  adopt  the  notation 

In  the  case  of  (4IO7),  the  equations  of  §726  and  727  give 


9..     /  ,,2  p2\  (^'         ~l)" 


^'  —  y  - 

27iR 


U, 


When  s^i  vanishes 

and    T  is  the  time  of  a  complete  revolution.     Wlien 

U,  =  n, 


—  421  — 

and  T  is    the    time    of  a   semi-revolution.     The   time  of  a  complete 
revoluiion,  ivhca  the  poidulum  moves  in  a  horkontal  circle  is 

so  ihat  it  is  proportional  to  the   square  root  of  the  distance  of  the  plane 
of  revolution  from  the  centre  of  the  sphere. 

731.      When   the   path   of  the   fendidiim   deviates    sligldtij  from    a 
horizontal  circle^  so  that  i  is  very  small,  the  notation 


^1  +  ^2  =  2^3=2  7^008(^3, 


gives 


«2  =  (;'  +  23)'-=— 3^— 


2-        *" 


A^  _  E-  —  zi 
¥g—~Tz'      ' 

—^ Z-C0S2  9, 

732.  AVhen  the  path  of  the  pendulum  deviates  slightly  from 
n  great  circle,  so  that  the  sum  of  z^  and  z.^  is  small,  p  is  large  and  i 
is  suKill,  the  formuli^  become,  by  neglecting  the  fourth  and  higher 
powers  of  / 


H.,-..)co.s2^  +  ^=±iI^%-= 


?/_  =  2  Jt 


—  422  — 

so    that   the   vibration   corresponds  to  a  complete   revolution   of  the  pen- 
dulum. 

733.  When  the  pendulum  passes  very  near  the  lower  point 
of  the  sphere,  so  that  Zi  differs  but  little  from  11,  the  neglect  of 
this  difference  and  its  higher  powers  gives 

2^:=.  R  COS  2  /, 

p:=z  Pi  -\-  tan^  ii^R  —  z-^^ 
_^_4^^(^-^0sin^/, 

z^  R  —  1R  sin^  i  sin^  cp  —  (7?  —  z^  cos^  ^^ , 
U^=.n  +  [cosec  ^  3^^ i  „)  _ if^'  S,  ( J  n)]  y'(2  -  ^)  ; 

so  that  the   vibration   corresponds  to  a  little   more   than  a  semi-revolution 
of  the  peiidtdiim. 

734.  In  the  general  case  the  vibration  of  the  pendulum  corresponds 
to  an  arc  of  revolution  which  exceeds  a  semi-revolution^  but  is  less  than 
an  entire  revolution.  When  the  velocity  at  the  highest  point  is  quite 
small,  the  case  of  §  733  occurs,  but  the  arc  of  revolution,  which  cor- 
responds to  a  vibration,  increases  with  the  increase  of  velocity  at 
the  highest  point.  When  the  highest  point  is  below  the  level  of 
the  centre  of  the  sphere,  the  case  of  §  731  gives  the  highest  limit 
of  the  velocity  at  this  point ;  but  when  the  highest  point  is  upon 
or  above  the  level  of  the  centre,  the  greatest  velocity  extends  to 
infinity,  which  limit  corresponds  to  the  case  of  §  732. 

735.  The  azimuth  of  the  pendulum  at  any  instant,  is  derived 
from  the  equation  of  §  726  in  a  form  suitable  for  computation  by 
means  of  the  following  formula? ; 

Zz:^  R  COS  ^^  , 

R  (cos  /9i  cos  /9._j  -|-  1 ) 


p  :=z  R  sec  a 


cos  di  -\-  COS  62 


—  423  — 
--  =  E"^  tan  a  ( cos  i\  -\-  cos  i\,)  = ,  - ,'    -    - , 

9   .  1  -\-  COS  «  COS  ^., 

COS"  I  =  — , 

1  -j-  cos  a  cos  Oi 

sin  -^  ^1 

.    •  cos  -A-  /9., 

,  1  -f-  COS  ^1  COS  6., 

tan  (/  —  — - — ~ — ' , 

sin  i7j  COS  a., 

.                                      ,    ,                 cos  6,  -\-  COS  ^., 
tan  //i  =;  cos  a  cos  tl,  tan  //  =  — V  /, ', 

'  sni  /^i  ' 

cos  /./'  =  COS  ^2  COS  fli , 

coi  =  i  n  —  i, 

^r^-^  I   _.  V  (cof^  qr  —  sin^  ?•  cos^  (jp) 
^^  cos  i  cos  fij  sin  d.j,        ' 

.  sin  M  tan  do  tan  4  0,  tan  1 

tan  ?.2  =  —     ,     \ — —^ , 

tan  c/j  cos  \i^ 
T  tan  ^. 

tan  /.o  =:  — s — Tv-i s-^ — ^-^  1 

tan-  ?/  ( 1  -|-  cos-  ^  tan-^  gi) 

^    tan''^  i  cos^  gi  cos  d.^  tan  j(<  tan  ^  Q^  tan  /, 

cos  j[i  cos  //  cos  /^,  -)-  (1  —  cos  II  cos  ?/  cos  ?/,)  sin^  /  sin-  g; ' 

^^  cos  ^  sin  ^ri  tan  ^]  r  ^A    /        cos- z  cos'- jr^  tan"^  i9.>       \         ~      1 

^^  —  tan  0,     ~     L     ''  \  tan^  \         " '  ^/  ~  '^'^  ^' J 

-|-  COS  i  COS  jW  tan  (^2^*^-1-  ^^i  —  ^^2  "h  ^-3  —  ^^4? 
and  the  arc  of  revolution  for  the  complete  vibration  is 

-\-  cos  i  cos  ,a  tan  ^2  9^y  ( i  ^ )  • 

These  formulae  do  not  appear  to  differ  from  those  of  Guuer- 
MANN,  although  the  reduction  is  more  extended.  They  give  with 
equal  fjicility  the  area  of  spherical  surface  which  is  described  by  the 
arc  of  a  great  circle,  wdiich  joins  the  extremity  of  the  pendulum  to 
the  lower  point  of  the  sphere. 


—  424  — 


MOTION    OF    A    FREE    POINT. 

736.  When  a  material  point  is  unconstrained  by  any  condi- 
tion, and  is  free  to  obey  the  action  of  any  force  whatever,  its 
motion  in  any  direction  is  simply  defined  by  the  equation 

737.  If  the  coordinates  are  assumed  to  be  of  the  partial  polar 
form  in  which 

z  =  the  distance  from  the  plane  of  .t\y , 

^)  =  the  distance  from  the  axis  of  .?, 

(^  =  the  inclination  of  ^  to  the  axis  of  .t, 

the  value  of  T  (1G228)  is,  for  the  unit  of  mass. 
The  corresponding  values  of  oj  (IGo^)  are 


(IJ  = 

*   J 

0)1=: 

0)2  = 

,/(f'; 

SO  that  0)2  25  the  double  of  the  projection  of  the  ladantaneous  area,  irhich 
is  described  bij  the  radius  vector  of  the  point,  upon  the  plane  of  x  y . 
The  equation  (I662)  gives,  then. 


n 


Tt  is  apparent  from  (oDu,)  that  ilie  second  mcmhcr  of  tJie  lad  equation 


.   —  425  — 

is  the  moment,   tidth   reference   to   the  axis  of  z,  of  all  the  forces  lohich 
act  upon  the  point. 

738.  If  the  forces  are  p-oportiomil  to  the  distances  from  the  centres 
from  u'hich  they  emanate,  the  hody  moves  as  if  it  ivere  under  the  influence 
of  a  single  force,  acting  hy  the  same  laiu  ivith  an  intensity  equal  to  the  sum 
of  the  intemities  of  the  given  forces,  and  emanating  from  a  centre  vMch  is  the 
centre  of  gravity  of  the  given  centres  regarded  as  masses  proportional  to  the 
intensities  of  their  action.  For,  if  the  notation  employed  in  §  128  is 
adopted,  and  if  m  denotes  the  sum  of  the  intensities  of  action,  the 
value  of  the  potential  is 


12 


J  ra 


in  which  ^  is  a  constant  and  can  be  absorbed  into  the  constant  II 
with  which  J2  is  connected  in  the  equations  of  motion. 

It  follows  from  §  685,  that  the  path  of  the  hody  is,  in  this  case, 
a  conic  section,  of  which  the  centre  of  gravity  is  the  centre. 

739.  If  all  the  forces  are  directed  towards  a  fixed  line,  tlie  area 
described  by  the  projection  of  the  radius  vector  upon  a  pUme  perpendic- 
ular to  the  fixed  line  is  proportional  to  the  time  of  desciipiion.  For  the 
instantaneous  area  is  in  this  case  constant  by  the  equation  (42429)  ?  in 
which  the  fixed  line  may  be  assumed  for  the  axis  of  ^,  so  that  the 
second  member  shall  vanish. 

740.  In  the  example  of  the  preceding  section,  a  peculiar  sys- 
tem of  coordinates  may  be  advantageously  adopted.  This  system 
consists  of  the  sum  of  the  distances  from  two  fixed  points  of  the 
given  line,  the  difference  of  these  distances,  and  the  angle  which 
is  made  by  a  plane  passing  through  a  fixed  line,  with  a  fixed  plane 
which  includes  this  line.     If,  then, 

2/  =  the  sum  of  the  distances  of  the  body  from  the  two  fixed 

points, 

54 


—  426  — 

2  5'  ^=  the  difference  of  these  distances, 
(f  =^  the  angle  which  the  plane  including  the  body  and  the 
fixed  line  makes  with  the  fixed  plane, 
2  «  =  the  distance  of  the  fixed  points  from  each  other, 
2  1/;  =  the  angle   which  the  two  lines,  which  are  drawn  from 
the  body  to  the  fixed  points,  make  wdth  each  other, 
k  =  the  perpendicular  drawn  from  the  body  to  the  fixed  lines, 

9  9  9 

9  9  9 

the  values  of  /c,yj,  and  T  are 

a   ' 

tan  11^  =  — , 

The    corresponding    differential    equations   derived   from    Le- 
grange's  canonical  forms  (164i2)  are 

i>,(^')  =  Z>,(Fy')  =  0, 


qpiqi''  I     ^qlpp'q'  -\-qi3\q 


qv  ^       '     ^  pl    '        «"       '  91 

The  integral  of  (4262o)  is 

in  which  B  is  arbitrary,  and  this  equation  expresses  the  proposition 
of  §739,  and  gives 

, £ Ba^ 

^         P        p'fql' 


—  427  — 

A   second   integral   of   these   equations,   corresponding  to  the 
principle  of  living  forces,  is 

The  sum  of  the  equations  obtained  by  multiplying  (42622)  by 
p\p',  (426^)  by  {-<ti'l)  and  (4273)  by  2  (pp'  +  qq')  is 

J  D,  [(/  - ./)  (/-  /)]  =  D.  [(/  +  rf-  «=)  (12  +  //)] 
-{</p'D,nJrp^q'D,P.). 

This  equation  is  integrable,  whenever  12  satisfies  the  condition 

2qD,n-2pD,n  =  (p^-f)Di,a, 

which,  by  the  substitution  of 

1  1 


may  be  transformed  into 

If,  then, 

—  —  (d  n       2<7-^\         A[(/— r)-Q] 

—  V      ''""         p-^  —  q^J—  f  —  (f  ' 

the  equation  (42Ti7)  oi"  (^^Tn)  becomes 

^^x'-D^  12,  +  Y'  D,  S2,  —  (x  +  ?/)  12, 
=  —  (l)    n    I     ^P-)=       I)^,[(p^--q^).Q]^ 


If,  then,  P  and  Q  are  arbitrary  functions  of  p  and  q  respective- 
ly, the  general  value  of  S2  is 


S2  = 


P±Q, 


—  428  — 

and,  if 

P?  =  2 II {f  _  «4^  -f  2  (P  +  C)  (/  —  d:-)  —  «2  i?2^ 
^2  =  2  H(q'  —  a')  4-  2  ( (2  —  (7)  («2  _  ^2)  _  ^^2^2^ 

in  which  C  is  an  arbitrary  constant,  the  integral  of  (4278)  is 
while  (4273)  may  be  written  in  the  form 

(/  -  q'f  {q\/+pl  /)  =  q\  p\+A  Ql 

It  is  easy  to  deduce  from  these  equations 

f  -=  f-^ 

J  p  ^i         J  q   ^i 

This  solution  is  published  by  me  in  Gould's  Astronomical  Journal. 

741.  It  is  evident  from  the  linear  form  of  the  equations 
with  reference  to  X2,  that  all  special  values  of  i2  may  be  combined 
into  a  more  general  value  by  addition  or  subtraction. 

742.  The  integrals  in  the  values  (428i5_2o)  assume  the  elliptic 
form,  when  P  and  Q  have  the  forms 

P  =  A,  +  Ap  +  Af  +  ^^+-^^, 

Q  =  B,  +  B,q  +  B,f  +  ^^^^  +  ^^, 
and  it  is  apparent  that,  in  the  expressions  of  the  integrals,  the  con- 


—  429  — 

stants,  Aq,  A2,  Bq  and  B2  can  be  combined  with  //,  0,  and  B.     The 
elliptic  integrals  become  circular,  when 

^=  —  A2^=^  B2, 

A,  =  B,  =  0, 

as  well  as  in  other  cases  which  do  not  seem  to  be  of  especial 
interest. 

743.  When  P  and  Q  have  the  forms 

F  =  A,p,     Q  =  B,q, 
the  value  of  the  potential  is 

so  that,  in  this  case,  the  forces  are  equivalent  to  tvjo  emanating  from  the 
fixed  points  ivith  the  same  laio  of  force  as  that  of  gravitation,  which  case 
has  been  integrated  by  Euler,  Lageange,  and  Jacobi,  and  the  forms 
of  Lagrange's  integrals  are  identical  with  those  of  (428ir_2o). 

744.  When  P  and  Q  have  the  forms 

P  =  Ap\      Q  =  —Aq\ 
the  value  of  the  potential  is 

SO  that,  in  this  case,  the  force  is  equivalent  to  a  single  force  emanating 
from  a  point  ivhich  is  midwag  hetiueen  the  two  fixed  points,  and  the  lavj 
of  force  is  proportional  to  the  distance  from  the  centime  of  force,  and  this 
case  is  integrated  by  Euler  and  Lagrange. 

745.  When  P  and  Q  have  the  form 

v\      p^ — ^^ '  ^1       ^^ — ?^ ' 


—  430  — 
the  value  of  the  potential  is 

•>       •'  ^^^        9    7  9  5 

piqi        a^¥' 

SO  that,  in  this  case,  the  force  is  equivalent  to  a  force  tvhich  emanates 
from  an  infinite  axis  of  uniform  extent,  and  is  inversely  proportional  to 
the  ciibe  of  the  distance  from  the  axis. 

746.  When  the  curve  is  given  upon  which  a  material  point 
moves  freely,  the  law  of  the  fixed  force  is  restricted  within  certain 
limits  which  it  may  be  interesting  to  investigate.  The  geometrical 
conditions  of  the  force  are  simply  that  it  must  be  directed  in  the 
osculating  plane  of  the  curve,  and  the  normal  force  must  be  equal 
to  the  centrifugal  force  of  the  body. 

By  the  adoption  of  the  notation 

the  equality  of  the  force  in  the  direction  of  the  normal  iV,  or  of 
the  radius  of  curvature  ()  to  the  centrifugal  force  is  expressed  by 
the  equation 

747.  Since  the  preceding  equation  is  linear,  all  the  special  values 
of  £2^  hj  tvhich  it  is  satisfied,  may  he  combined  into  a  neiv  value  hy  ad- 
dition or  subtraction.  Previously  to  this  addition,  each  value  of  il^  may 
he  multiplied  hy  a  factor,  which  may  represent  the  mass  of  the  body, 
and  if  the  factor  is  denoted  by  m,  the  value  of  m  il^  will  correspond 
to  the  whole  force  acting  ujDon  the  mass,  and  it  is,  then,  evident 
that,  if  M  denotes  the  mass  upon  which  the  combined  forces  act, 
and  V  its  velocity,  the  combined  power  is 

MV'=:Eiymv^), 
which  expresses  a  condition  identical  mth  the  theorem  of  Bonnet. 


—  431  — 

748.  If  a  special  value  of  12^  is  represented  by  /2o,  and  if  £1^ 
satisfies  the  equation 

so  that  it  is  the  potential  of  a  force,  to  the  level  surfaces  of  ^vliich 
the  given  curve  is  a  perpendicular  trajectory,  the  complete  value  of 
i2i  is 

ill  ivliich  f  is  an  arhitrarjj  function.  It  is  apparent,  then,  that  D.^  has 
an  endless  variety  of  possible  forms  in  every  special  case.  But 
each  form  corresponds  to  an  arbitrary  value  of  one  of  the  constants 
of  the  given  curve,  or  of  some  combination  of  those  constants. 

749.  If  the  given  curve  is  the  parabola,  of  which  the  equa- 
tion is 

the  values,  which  correspond  to  the  arbitrary  value  of  .Tq,  are 

/22==log(.y  — ,yo)  + 


X 


(1.= 


while  those,  which  correspond  to  the  arbitrary  value  of  ^u,  arc 


f2o=2(.r-.r„)+|;; 


and  it  is  interesting  to  observe  that  when,  in  this  case,  the  arbitrary 
function  of  i22  is  assumed  to  be  constant,  the  value  of  the  force  is 
independent  of  x^^  and  p  as  well  as  of  ^o- 

The  values,  which  correspond  to  the  arbitrary  value  of  ^j,  are 


—  432  — 

750.     If  the  given  curve  is  the  conic  section,  of  which   the 
equation  is 

ecos{(p  —  cpQ)  =  -  —  l, 
the  values,  which  correspond  to  arbitrary  values  of  (p^  are 


r    ^ 


■0 


in  which 

B^  =  (^r'  —  {Fr  —  9^f. 

When  the  arbitrary  function  of  SI^  is  assumed  to  be  constant, 
the  force  is  independent  of  e  and  P  as  well  as  of  (pQ,  and  its  law  is 
identical  with  that  of  gravitation. 

751.  If  the  given  curve  is  the  cycloid  determined  by  the 
equations 

^— ^0  =  ^(1  — cos^), 
X  —  .To  ^^  Ji{^  —  sin  t5) ; 

the  values  which  correspond  to  arbitrary  values  of  a-Q  are 

^2  =  ^'  +  ^  (^  +  sin  &), 
1 


^2o  = 


y—yo' 


in  which  ^  is  to  be  regarded  as  the  function  of  ?/,  which  is  deter- 
mined by  (432i8). 

752.  If  the  given  curve  is  a  circle  of  tvhich  the  centre  is  the  ori- 
gin ivJiile  the  radius  is  arUtrarg,  the  potential  of  the  force  is  an  arUtrary 
homogeneous  function  of  the  reciprocals  of  x  and  y,  tvhich  is  of  the 
second  degree. 


—  433  — 

This  peculiar  result  is  the  more  worthy  of  attention  because 
it  can  be  extended  to  the  sphere,  so  that  the  iiotential  of  a  force  hy 
tvhich  a  body  may  move  upon  a  sphere  of  a  given  centre  hut  of  an  ar- 
bitrary radius,  is  Ukeivise  an  arbitrary  homogeneous  function  of  the  second 
degree  of  the  reciprocals  of  the  rectangular  coordinates,  of  tvhich  the  centre 
of  the  sphere  is  the  origin. 

These  problems  are  fruitful  of  new  subjects  of  interesting  geo- 
metric speculation. 


CHAPTER    XII. 

MOTIOX    OF    ROTATION. 

753.  If  the  coordinates  of  the  points  of  a  sj^stem  are  the 
partial  polar  coordinates  of  §  737,  and  if  (p^  is  supposed  to  refer  to 
some  point  of  the  system,  that  is,  to  an  axis  connected  with  the 
system,  from  which  the  corresponding  angles  ^  are  measured,  so 
that  the  value  of  (f  is 

9  =  To  +  ^. 
that  of  T  becomes 

T=h^,[ln,{:4  +  ^  +  ^^^!^)]• 
Hence  the  equation  (164i2)  gives 

D,n  =  D,2,{m,q-',if\), 

the  second  member  of  which  is  the  derivative  of  doul)le  the  sum  of 
the  products  obtained  by  multiplying  each  element  of  mass  by  the 

55 


—  434  — 

area  described  by  the  projection  of  the  radius  vector  upon  the 
plane  perpendicular  to  the  axis  of  rotation.  If  this  area  is  desig- 
nated as  the  rotation-area  for  the  axis,  it  follows  from  (oOgs)  that  the 
derivative  of  the  rotation-area  for  the  axis  is  equal  to  the  sum  of  the 
moments  of  the  forces  loitli  reference  to  that  axis.  It  is  obvious  that 
the  mutual  actions  of  the  system  may  be  neglected  in  obtaining 
the  sum  of  the  moments. 

If,  then,  all  the  external  forces  tvhich  act  xqion  a  system  are  directed 
towards  an  axis,  the  rotation-area  for  that  axis  tvill  he  described  with  a  uni- 
form motion,  which  is  the  principle  of  the  Conservation  of  Areas. 

754.  The  rotation-area  for  an  axis  may  be  exhibited  geomet- 
rically by  a  portion  of  the  axis  which  is  taken  proportional  to  the 
area,  and  it  is  evident  from  the  theory  of  projections  that  rotation- 
areas  for  different  axes  may  be  combined  by  the  same  laws  with 
which  forces  applied  to  a  point,  and  rotations  are  combined,  so  that 
there  is  a  corresponding  parallelopiped  of  rotation-areas.  There  is,  then, 
for  every  system  an  axis  of  resultant  rotation-area,  tvith  reference  to  which 
the  rotation  is  a  maximum,  and  the  rotation-area  for  any  other  axis  is  the 
corresponding  projection  of  the  resultant  rotation-area.  The  rotation-area 
vanishes,  therefore,  for  an  axis  zvhich  is  perpendicular  to  the  axis  of 
resiUtant  rotation-area. 


ROTATION     OF     A     SOLID     BODY. 

755.  In  the  rotation  of  a  solid  body,  the  axis  of  rotation  does 
not  usually  coincide  with  that  of  resultant  or  maximum  rotation- 
area  ;  and  the  relations  of  these  two  axes  is  of  fundamental  impor- 
tance in  the  investigation  of  the  rotation.  The  determination  of 
these  relations  depends  directly  upon  the  moment  of  inertia.  The 
moment  of  inertia  of  a  hody  or  system  of  bodies  upon  an  axis  is  the  snni 


—  435  — 

of  the  products  obtained  hj  midtiplying  each  element  of  mass  hy  the  square 
of  Us  distance  from  the  axis. 

The  distorting  moment  with  reference  to  two  rectangular  axes  is  the 
sum  of  the  products  obtained  by  midtiplying  each  element  of  mass  by  the 
pivducts  of  its  distances  from  the  two  corresponding  coordinate  planes. 

Let  then 

9n  =  the  mass  of  the  body, 

(^j^^=r=the  distance  of  the  element  dm  from  the  axis  of^:*,  which 
passes  through  the  origin, 

-  Ij'f=^  the  reciprocal  of  the  moment  of  inertia  for  the  axis  of  p, 

ni  J.,  =  the  distorting;  motion  of  inertia  for  the  two  axes  which  form 
a  rectangular  system  with  the  axis  of  p, 

which  gives 

^,.  J  in 

If,  then,  ^)^  is  the  angle  which  the  axis  of  p  makes  with  the  direc- 
tion of  q,  the  moment  divided  by  the  mass,  becomes 

5 = ^^X  (''  '"'"^  ^^)  ^  ^^  X  (^'  - "' '"''  ^^) ' 

—  ^.,  (^^^^  —  2  j;  cos  (fy  cos  9),) . 

If  Ij,  is  set  off  upon  the  axis  from  the  origin,  its  extremity 
lies  upon  a  finite  surface  of  the  second  degree,  which  is,  therefore, 
an  ellipsoid,  and  may  be  called  the  inverse  ellipsoid  of  inertia.  If  the 
axes  of  this  ellipsoid  are  assumed  for  the  axes  of  coordinates,  the 
values  of  J  must  vanish  for  each  of  these  axes,  that  is,  there  is  no 


—  436  — 

distorting  inertia  for  these  axes  which  mafj  he  called  the  principal  axes  of 
inertia. 

756.  When  a  body  rotates  about  an  axis,  the  rotation-area 
for  an  axis,  which  is  perpendicular  to  that  of  rotation,  is  obviously 
proportional  to  the  distorting  inertia  for  these  two  axes.  There  is, 
therefore.,  no  rotation-area  for  a  principal  axis  of  inertia  proceeding  from 
rotation  about  either  of  the  other  ttvo  axes  of  inertia. 

757.  If  ^p  is  the  velocity  of  rotation  about  the  axis  of />,  the 
corresponding  velocity  of  rotation  about  the  principal  axis  of  x  is 

(3;=:^;cosf/)^, 

and  the  corresponding  rotation-area  is 

m  6'j,  cos  (px 

the  cosines  of  the  angles,  which  the  axis  of  resultant  rotation-area 
makes  with  the  principal  axes,  are  then  proportional  to 

COSJp^    COSJP^      ,  COSJPj 

■'■X  -*!/  -^Z 

SO  that  this  axis  coincides  with  the  perpendicular  to  the  tangent 
plane  of  the  ellipsoid  which  is  drawn  at  the  extremity  of  the  axis 
of  rotation.  The  plane  of  niaxinium  rotation-area  is,  therefore,  conjugate 
to  the  diameter  of  the  ellipsoid  tuhich  is  the  axis  of  rotation,  which  theorem 
is  given  by  Poinsot. 

758.  If  the  reciprocal  of  the  perpendicular  let  fall  from 
the  origin  upon  the  tangent  plane  of  the  ellipsoid  is  set  off  upon 
the  perpendicular,  its  extremity  lies  upon  a  second  ellipsoid,  which 
may  be  called  the  ellipsoid  of  inertia,  and  of  which  the  principal  axes 
are  the  reciprocals  of  the  principal  axes  of  the  ellipsoid  of  §  755,  and  are 
proportional  to  the  square  root  of  the  principal  moments  of  inertia. 

759.  It  is  apparent //iff?^  the  tangent  plane  to  the  ellipsoid  of  inertia 


—  437  — 

ivhich  is  draivn  at  the  extremiti/  of  the  axis  of  maxinmm  rotation-area  is 
perpendicular  to  the  axis  of  rotation. 

It  is  also  evident,  that  the  axis  of  rotation  is  one  of  the  principal 
axes  of  the  section  of  the  inverse  ellipsoid  of  inertia,  tvhich  is  made  hy  a 
plane  passing  thro%igh  the  axis  of  inertia  and  perpendicular  to  the  common 
plane  of  the  axis  of  rotation  and  of  maximum  rotation-area,  tvhile  the  latter 
axis  is  one  of  the  principal  axes  of  the  section  of  the  ellipsoid  of  inertia, 
tvhich  is  made  hj  a  passing  plane  through  this  axis  perpendicular  to  this 
same  common  jjlane. 

760.  Although  the  fixed  axes  of  coordinates  may  be  assumed 
at  any  instant  to  coincide  with  the  principal  axes  of  inertia,  the  axes 
of  inertia  are  nevertheless  in  constant  motion  from  the  fixed  axes, 
and  at  the  end  of  the  instant  dt,  after  coincidence,  the  axes  of  rota- 
tion, which  coincided  at  the  beo-innino;  of  the  instant  with  the  fixed 
axes  of  g  and  z,  will  not  remain  perpendicular  to  the  fixed  axis 
o^  X,  but  will  deviate  from  perpendicularity  by  the  respective  angles 

^'^dt  and  — ^'y  dt. 

The  rate  of  increase  of  the  rotation-area  for  the  fixed  axis  of  x, 
which  arises  from  the  external  forces  is,  therefore, 


m    ^x  n         "  y  \u     L-f 


which  represents  the  ivell-known  equations  given  hy  Euler  for  the  rota- 
tion of  a  solid  body. 

If  the  rotation-area  for  the  axis  of  jt?  is  denoted  by  m  A^^,,  the 
preceding  equation  may  assume  the  form 

lD,a=.D,K~  A',  a:  (H-  /;) . 

761.     If  the  equation  (43722)  is  multiplied  by  2  ^'^  and  added 


—  438  — 

to  the  corresponding  products  for  the  other  axes,  the  integral  of 
the  sum  is 

m  "^  I^^ 

which  is  simply  tJie  equation  of  living  forces.  If  p  is  the  semidiame- 
ter  of  the  inverse  ellipsoid  of  inertia,  about  which  the  solid  is 
revolving  at  the   instant,  the  preceding  equation  may  be  reduced  to 


ROTATION    OF    A    SOLID    BODY    WHICH    IS    SUBJECT    TO    NO    EXTERNAL    ACTION. 

762.  If  a  solid  body  is  subject  to  no  external  force,  the  centre 
of  gravity  may  be  assumed  for  the  origin.  In  this  case  the  first 
member  of  (437.22)  oi'  (43729)  vanishes,  and  the  equation  (438g) 
becomes 

2)  m 

or 

so  that  the  velocity  of  rotation  is  proportional  to  the  diameter  of  the  inverse 
ellipsoid  which  is  the  axis  of  instantaneous  rotation,  which  is   given   by 

POINSOT. 

763.  It  follows  from  §§757  and  762,  that,  if  q  is  the  perpen- 
dicular let  fall  upon  the  tangent  plane  which  is  drawn  to  the 
inverse  ellipsoid  at  the  extremit}^  of  the  axis  of  rotation,  q  is  the 
axis  of  maximum  rotation-area,  which  is  invariable  when  there  is 
no  external  force,  and  that 


—  439  — 

^',  =  &'p  COS  ^)p  =  hjl  COS  (fp  ^hq=.  -', 

SO  that  the  velocity  of  rotation  about  the  axis  of  maximum  rotation-area^ 
as  ivell  as  the  distance  of  the  tangent  plane  vjhich  is  draivn  to  the  inverse 
ellipsoid  of  inertia  at  the  extremity  of  the  axis  of  rotation  are  invariable 
dming  the  motion  of  the  solid^  which  are  propositions  given  by  Poinsot. 
They  might  have  been  deduced  ^yith  faciHty  from  the  geometrical 
theorem  of  §  759,  without  the  aid  of  the  equation  of  Hving  forces, 
which  might  on  the  contrary  have  been  derived,  in  the  present  case, 
as  an  inference  from  these  theorems,  and  this  was  the  elegant  pro- 
cess of  Poinsot. 

If  the  solid  body  has  no  translation,  the  inverse  ellipsoid  re- 
mains constantly  tangent  to  the  same  plane  which  is  that  of  max- 
imum rotation-area,  and  which  touches  the  ellipsoid  at  the  extremity 
of  the  axis  of  rotation.  It  is  apparent,  then,  that  in  the  motion  of 
the  solid,  the  ellipsoid  rolls  upon  the  fixed  plane  of  maximum  rotation-area, 
uithout  awj  sliding ;  which  is  Poixsot's  mode  of  conceiving  this 
motion. 

764.  The  instantaneous  axis  moves  within  the  body  in  such  a  umg 
as  to  describe  the  surface  of  the  cone  of  the  second  degree,  of  Avhich  the 
equation  is 


-•n{h-m=^- 


The  base  of  this  cone  is  an  ellipse  perpendicular  to  the  greatest  axis  of 
the  inverse  ellipsoid  ivhen  q  is  larger  than  the  middle  axis,  or  impen- 
dicular  to  the  least  axis,  when  q  is  less  thrrn  the  middle  axis;  and  in 
either  case  the  centre  of  the  ellipse  is  upon  the  axis  to  which  it  is  per- 
pendicular. 

When  q  is  equal   to  either  the  greatest  or  the  least  axis,  this 
axis  becomes  the  permanent  axis  of  rotation  ;  but  when  q  is  equal 


—  440  — 

to  the  middle  axis,  the  cone  is  reduced  to  a  plane  which  corresponds 
to  one  of  the  plane  circular  sections  of  the  ellipsoid  of  inertia. 

The  axis  of  maximum  rotation-area  describes  within  the  body 
the  cone  of  the  second  degree  of  which  the  equation  is 

The  common  plane  of  the  instantaneous  axis  of  rotation  and  of  the 
axis  of  maximum  rotation-area  is  ohviously  normal  to  the  surface  of  the 
cone  described  in  the  hody  hy  the  axis  of  maximum  rotation-area,  which 
defines  the  relative  position  of  these  two  axes  at  each  instant. 

765.  The  position  of  the  axis  of  maximum  rotation-area  is 
fixed  in  space,  and,  therefore,  the  path  of  the  instantaneous  axis 
of  rotation  in  space  is  determined  by  the  preceding  property,  and  a 
distinct  geometrical  idea  of  the  cone  described  by  the  instantaneous 
axis  in  space,  is  obtained  by  conceiving  the  cone  described  in  the 
body  by  the  axis  of  maximum  rotation-area  to  be  compressed  into 
a  line  carrying  with  it  the  cone  described  by  the  instantaneous  axis, 
in  such  a  way  as  not  to  change  the  relative  inclination  of  the  two 
axes  or  the  surface  of  the  cone  of  the  instantaneous  axis. 

The  algebraic  definition  of  the  cone  of  the  instantaneous  axis 
in  space  is  obtained  by  assuming  the  axes  of  the  inverse  ellipsoid 
to  be  arranged  in  the  order  of  magnitude  as  x,  y,  z,  in  which 
the  cone  of  the  axis  of  rotation  has  the  axis  of  x  as  its  central  axis, 
and  adopting  the  notation 

cos  E^^=^ , 


Ml 


=  ^  +  ^-^?  =  ^^-^(l-f)(l-f): 


and  similar  equations  for  the  other  axes,  in  which  it  is  unimportant 
that  the   angles  E-,  may  be  imaginary,  but  it  should  be   observed 


—  441  — 

that  i)[fy  is  the  largest  of  the  quantities,  M^,  3Iy,  and  M^;  the  fol- 
lowing notation  is  also  to  be  adopted. 

sin^  Ez=  —  sin^  E^  sin^  Ey  sin^  E^ , 

^=(i-f)(i-f)(i-|) 

=  N/[(i-f)(i-f)(i-f)]. 

J.Uy 

±J±y 

sin  t] 

sm « =  -. — -, , 

sin  1] 
^  J/^  sin  ;/ 

sin  (5  =  sin  2  sin  ^  ; 

which  give  for  each  axis,  if  x,i/,z  denote  the  extremity  of  the  in- 
stantaneous axis  upon  the  surface  of  the  inverse  ellipsoid, 

7,2  ^  sin^  E=  I^  (/  —  M!)  sin^  E^, 

D  X p  Dp 

~^~  f  —  M'i' 

I^  Dx'  sin^  E^  =  f^^^^^Df^ 
p^  —  Mi 

If,  then,  If  is  the  angle  which  the  plane  of  p  and  q  makes 
with  a  fixed  plane  passing  through  ^,  the  cone  of  the  instantaneous 
axis  in  sj)ace  is  defined  by  the  equation 

f  sin^  y^  D  v^2  -^  _Z_=  />.^_|_  z)^2_|_  2>  ^2^ 
or 

^<pW  —  (^2_^.) ^i^p^-Mi) {m;--p^)  {:p'-m^)\ 

9'  sec  (9     1^  {<f  -\-  M^  —  M'^  sin^  t]  sin^  (f)  My  sin ;/'  _ 

My  sin  //  "■"  9^  Nsec  d  ' 

5G 


—  442  — 
which  gives  by  the  use  of  elhptic  integrals 

766.  The  velocity  with  which  the  instantaneous  axis  moves 
in  the  body  is  readily  obtained  from  the  equations  of  the  preced- 
ing section,  which  give  in  combination  with  (43722) 

pD,})  =  :S^  {x  A  x)  —  —hxijz^^ ^ — :^^ 

■^X    -*!/    -'Z 

whence 


and  by  elliptic  integrals 

My  sin  r(  h  {t  —  r )  =  ^^  9) . 
767.     In  the  especial  case  of 

the  axis  of  maximum  rotation-area  describes  one  of  the  circular  sections  of 
the  ellipsoid  of  inertia,  and  the  equations  of  §  765  become 

i¥,  =  iK  =  7,, 

cos  7]  =:  cos  7]   =2—^, 

V 

iz=.  ^  n, 
hly{t  —  t)  =^s 
y/  (31^  — 7/)  =  3fy  sin  t]  Tan  [/^  (^f  —  t)  My  sin  ij] 

=  v^  {31^  —  If  sec^  cp,)  =  My  sin  t^  Tan  (t^Lipi) . 


—  443  — 

The  greatest  value  o?  p  is,  then,  3I,j,  which  corresponds  to 

t  —  T  =  o  ; 
and  its  least  value  is  ly,  which  corresponds  to 

t  —  T  =  =b  GO  . 

If  the  axis  of  rotation,  therefore,  coincides  with  the  mean  axis  of  the 
ellipsoid  of  inertia  at  the  commencement  of  the  motion,  its  position  ivill  l)e 
permanent  in  the  ellipsoid,  although  it  is  affected  ivith  an  element  of  insta- 
Ulity  ;  hut,  in  all  other  instances  of  the  present  case,  the  axis  of  rotation 
describes  the  spiral  of  which  (4423o)  is  the  ecpiation,  and  is  constantly  ap- 
proaching the  mean  axis  at  such  a  diminishing  rate  of  velocity  that  it  never 
reaches  this  axis. 

768.  When  the  ellipsoid  of  inertia  is  one  of  revolution,  the 
cones,  described  by  the  instantaneous  axis  in  the  body  and  in  space, 
are  both  of  them  cones  of  revolution,  so  that  the  simplicity  of 
this  case  requires  no  further  illustration ;  but  it  may  be  ob- 
served, that,  when  the  ellipsoid  is  oblate,  the  moving  cone  rolls 
externally  upon  the  stationary,  but  internally  when  the  ellipsoid 
is  prolate. 

769.  This  analysis,  which  is  substantially  the  same  with  one 
of  the  forms  of  Polxsot,  comprehends  the  principal  conclusions  of 
EuLER,  Lagrange,  and  Laplace,  and  may  be  extended  to  the  case  in 
which  the  origin  is  any  fixed  point  of  the  solid. 


THE     GYROSCOPE     AND     THE     TOP. 

770.  When  the  solid  is  subject  to  any  accelerated  force,  and 
its  gyration  is  about  a  fixed  point,  which  may  be  assumed  as  the 
origin,  and  when  the   ellipsoid  of  inertia  with   reference    to    this 


—  444  — 

point  is  an  ellipsoid  of  revolution  about  the  axis  of  *,  the  corre- 
sponding Eulerian  equation  is 

771.  If  the  body  is  also  symmetrical  about  the  axis  of  z,  the 
preceding  equation  becomes 

^',  =  n, 

so  tJiat  the  rotation  about  the  axis  of  z  is  uniform. 

772.  If  the  force  is  that  of  gravitation,  the  problem  becomes 
that  of  the  gyroscope.  If  g  is  the  direction  of  gravitation,  h  that 
of  a  horizontal  axis,  which  is  perpendicular  to  the  axis  of  0,  and 
has  that  direction  about  which  the  rotation  from  ^  to  ^  is  posi- 
tive, if 

I  —  ^ 
s  —  2) 

/   X) 

if  /  is  the  distance  of  the  centre  of  gravity  from  the  origin,  l^  the 
projection  of  /  upon  g ^  and  if  the  gyration  of  the  body  is  resolved 
♦  into  the  three  rotations,  /'  about  the  axis  of  ^,  |'  about  the  axis 
of  h,  and  v^i'  about  the  axis  of  g^  the  rotation  about  the  prin- 
cipal axes  are 

(5;  =  1//  cos  1 4- '/!, 

(5;  =  if/cosf  +^"cos/, 
^'y  =  1//  cos  I  —  r  sin  / . 

These  equations  give 

^'x  COS  I  -|-  ^'y  COS  %  =  ^''  sin^  l^\\>'  \\  —  ^. 


—  445  — 

Tlie  area  about  the  axis  of  g  is  evidently  described  uniformly 
by  the  principles  of  §  753,  so  that  if 

a  =  y  y/  /, 

and  if  ^  is  a  constant,  we  have 

l'^  (^^  cos  %  -f"  ^'y  cos  %)  -\-  a- 1  ^'^  cos  f  =  (/^  —  /|)  I//  +  n  i^  ^  =^  '^  ^^''  4 • 
The  equation  (4883)  gives,  in  the  present  case, 

(/2  _  ^1)2  ^/^^  ^2  ^^^^  ;,2  (^2  _  ^1)  (^^^  _  ^^)  ^ 

provided  the  constants  /^  and  4  ♦'ii'©  determined  by  the  equations 

The  elimination  of  i//  from  the  equations  (445;)  and  (445io) 
gives 

^2  /;'  =:  7,2  (/2  _  ^^  (/^  _  ^^)  _  ^,2  ^^4  (^^  _  ^J2_ 

773.  The  limiting  values  of  4  correspond  to  the  vanishing 
values  of  /^,  and,  therefore,  reduce  the  second  member  of  (445i9)  to 
zero.  If  these  values  are  denoted  by  4,  4?  and  — p,  it  is  evident, 
from  the  form  of  the  equation,  that  p  is  greater  than  I,  while  4  and 
4  are  included  between  — /and  -|-/.  The  equations  for  the  sjDher- 
ical  pendulum  of  §§  726  and  727  may  be  directly  applied  to  the 
gyroscope  by  changing  z  into  4  and  *o?  ^1?  ^'^^  ^2  iiito  4>  A?  ^ii^  4? 
which  give,  by  (41824-27)? 


—  446  — 

With  this  notation  and  the  equations  derived  from  (418i3_i4)5  the 
expression  of  the  time  is 

and  the  equation  of  the  path  described  by  the  axis  of  this  body 
in  sjDace  is 

which  admits  of  reduction  by  the  process  of  §  735. 

774.  When  the  velocity  n  vanishes,  the  gyroscope  is  re- 
duced to  a  case  of  the  spherical  pendulum  of  which  the  length  is 

775.  When  the  two  roots  /^  and  4  are  equal,  the  path  of  the 
gyroscope  k  a  horizontal  circle.  The  values  of  4?  <iiitl  of  the  velocity 
of  rotation  can  be  determined  for  this  case  by  the  equations 

The  denominator  of  the  value  of  (/i  —  4)  can  be  written  in  the  form 

^ '  +  A  4  —  3  /i^  =  3  ( 4  —  /, )  ( /:  +  4 ) , 

in  which  4  ^^^  4  ^^'^  positive  quantities.  If,  then.  4  is  greater 
than  4 J  ^''  is  positive;  but  when  4  is  contained  between  4  and  4?  w' 
is  negative  ;  and  4  can  never  be  contained  between  —  4  and  —  /. 


—  447  — 

776.     When  the  values  of  4  and  l^  are  equal,  \\i'  vanishes  at 
the  same  time  with  t',  and  we  shall  also  find 

4  ^=  <  1  ^=  U  J 
and  the  equation  for  determining  4  and  p  is 

This  is,  approximately,  the  ordinary  case  of  the  gyroscope,  and  it  is 
evident  that  in  this  case  the  values  of  k  and  4  cannot  be  equal, 
unless 

so  that  the  centre  of  the  gyroscope  cannot  wider  these  circumstances  de- 
scribe a  horizontal  circle,  which  coincides  with  the  conclusion  of  Major 
T.  G.  Barnard. 

J^,  hmvever,  in  this  case,  n  is  very  large,  it  is  obvious  that  the  differ- 
ence between  \  and  \  is  quite  small,  for  this  difference  is 

which  is  also  one  of  the  results  obtained  by  Major  Barnard. 

777.  When  4  is  algebraically  greater  than  4,  it  is  also  alge- 
braically included  between  4  and  4,  so  that  \\'  is  positive  at  the 
upper  limit,  and  negative  at  the  lower  limit.  But  when  4  is  alge- 
braically smaller  than  4,  it  is  also  algebraically  smaller  than  either 
4  or  4,  so  that,  in  this  case,  \^'  is  always  negative. 

778.  When  4  is  equal  to  /,  it  is  also  equal  to  4,  that  is, 

4  =  /  =  4. 

The  velocity  of  rotation  which  corresponds  to   this  case  is  deter- 
mined by  the  equation 

n^_  (H-3)  Q-2-z}^  _  (!'—v)iv^3 


—  448 
which  gives 


I> 


_         2l(k  —  Q 
—  ^"T"      l-l,     ' 


^^^'  —  2i(i-i,y 
^,  _ ,  /  r(/+4)(4-Q]  op  (^11^  A 

^'  —  \  L     2l{l-l,)     J  -^'V    2/  '^Z- 

779.  When  4  is  equal  to  — /,  it  is  also  equal  to  4,  that  is, 

In  this  case  4  is  algebraically  less  than  —  /,  and  the  velocity 
of  rotation  which  corresponds  to  this  case  is  given  by  the  equation 

which  gives 

2l(I  +  Q 

780.  When  p  is  equal  to  — /,  it  is  also  equal  to  4,  that  is, 

In  this  case  4  is  algebraically  greater  than  —  /,  and  the  veloc- 
ity of  rotation  which  corresponds  to  this  case  is  given  by  the 
equation 


which  gives 


n^ _  (I -I,)  (I,-!,)  _  (l-l^  (1,-1,) 
/r  l-{-li  Z4-4  '■ 


2I(I+h) 

^2— -qi^^       i' 


—  449  — 

781.  If,  in  the  preceding  case,  4  is  equal  to  the  negative  of 
/,  it  will  also  be  equal  to  4?  that  is, 

and,  in  this  case,  the  elliptic  integrals  disappear  from  the  equations, 
so  that  they  become 

4  =  /,-(4  +  /)Tan^[A(^-T)v/(/+/i)], 

and  although  the  axis  is  constantly  approaching  the  upper  veriical,  after 
passing  the  loiuer  limit,  it  never  reaches  the  upper  limit ;  and  if  it  hegiiis 
at  the  upper  limit  it  never  recedes  from  it. 

782.  In  the  simplest  form  of  the  problem  of  tlie  spinning  of  the 
top,  the  extremity  of  the  hody  is  a  point  in  the  axis  of  revolution,  tuhich 
is  restricted  to  move,  vMhoxit  friction,  in  a  horizontal  plane.  In  this  case, 
the  equation  (4449)  is  still  applicable,  as  weU  as  (4467),  provided 
that  the  moments  of  inertia  are  referred  to  the  centre  of  gravity 
of  the  top,  and  that  I  denotes  the  distance  from  the  centre  of  grav- 
ity to  the  point  in  the  horizontal  plane. 

The  equation  (4883)  gives,  in  this  case,  with  the  notation  of 
§772, 

{p-ll)v^'  -^l\\^P n-I^ll)Vr^lr{p-n){k-h)\ 

and  if  w'  is  eliminated  by  means  of  (445;), 

The  comparison  of  this  equation  with  (445i9),  shows  that  the 
limits  of  motion  are  the  same  as  in  the  case  of  the  gyroscope,  and 

57 


—  450  — 

under  the  condition  of  the  equality  of  l^  «ind  4?  the  extremity  of 
the  axis  of  the  body  describes  a  horizontal  circle.  The  expressions 
of  the  time  and  of  the  azimuth  of  the  axis  are  not,  however,  capable 
of  expression  by  means  of  elliptic  integrals,  except  in  special  cases, 
of  which  that  of  §  781  is  one,  and  another  corresponds  to  the  case 
of 

■'-X 

783.  When  the  horizontal  plane,  to  which  the  extremity  of 
the  top  is  restricted,  is  not  smooth,  the  problem  is  usually  more 
complicated,  although  when  the  friction  brings  the  lower  extremity 
to  the  case  of  rest,  it  reassumes  the  form  of  the  gyroscope,  and  this 
is  the  modification  of  the  problem  which  has  been  investigated  by 
PoissoN.  In  this  case  of  the  gyroscope,  hoivever,  the  friction  becomes  an 
interesting  feature  of  the  proUem,  and  has  a  peculiar  effect  upon  the 
limits  to  which  the  motion  is  subjected.  Instead  of  the  equation  (4449), 
the  rotation  about  the  axis  of  the  body  decreases  uniformly,  which  is  ex- 
pressed by  the  equation 

The  area  described  about  the  vertical  axis  is  also  described  in  this 
case,  at  a  uniformly  decreasing  rate,  which  gives  instead  of  (4467), 

{p  —  li)y\>'  +  {n  —  ^n)a-l^^n(^-{k  —  t^t). 

The  power  of  the  system  is  reduced  by  the  friction  about  the 
body-axis,  which  is  proportional  to  the  angle  1,  and  by  the  friction 
about  the  vertical  axis,  which  is  proportional  to  \\^ .  If,  then,  the 
mean  values  of  i//  and  ^  for  a  small  interval  of  time  are  denoted  by 
T/V  ^ncl  ^„j,  the  equation  of  the  preservation  of  power  may  be  re- 
duced to 

{P  -  P,)  v/^  +  (^  if  =  If  (P  -  li)  (/^  +  /„  +  /„ 0 , 


451 


in  which 


2i7«<=|J(V''cos?)-"^> 

n'      ,    .          ..  n  a^Vo     ,    , 

=  j^XjJ^t  COS  ^^ -JYJ2  Wmt- 

The  combination  of  this  equation  with  (45O23)  gives 

It  is  obvious  from  this  equation,  that  if  the  friction  about  the 
body-axis  vanishes,"  the  height,  to  w^hich  the  gyroscope  ascends, 
diminishes  at  each  oscillation.  If,  however,  the  friction  about  the 
vertical  axis  is  destroyed,  the  height,  to  which  the  gyroscope 
ascends  at  each  oscillation,  increases  when  the  body-axis  is  directed 
upwards  in  its  mean  position  ;  but  this  height  diminishes  when  it 
corresponds  to  a  position  in  which  the  centre  of  gravity  is  below 
the  fixed  extremity  of  the  axis.  In  all  intermediate  positions,  and 
when  both  the  frictions  remain,  the  increase  or  decrease  of  ascent 
depends  upon  the  peculiar  relations  of  the  various  constants. 

In  the  spinning  of  the  top,  the  rounded  point  rolls  upon  the 
suj^porting  plane,  which  induces  an  acceleration  about  the  vertical 
axis  which  is  the  reverse  of  friction,  and  this  is  the  principal  cause 
of  the  ready  rising  of  a  top  into  the  vertical  position  of  apparent 
repose,  known  as  the  sleeping  of  the  top. 


THK    DEVIL    ox    TAVO     STICKS    AND    THE    CHILD's    HOOP. 

784.  Contrasted  with  the  motion  of  the  gyroscope  is  that 
of  a  solid  of  revolution  of  which,  instead  of  a  fixed  point  of  the 
axis,  the   circumference  of  a  section  drawn  through  the  centre  of 


—  45'2  — 

gravity,  perpendicular  to  the  axis  is  restricted  to  move  upon  a 
point.  A  convenient  type  of  this  class  of  motion  may  be  found 
in  the  familiar  toy  called  the  devil  on  tivo  sticks.  If  the  friction  is 
neglected  in  this  case,  and  the  notation  adopted  from  the  preceding 
problem  of  the  gyroscope,  the  rotation  about  the  body-axis  is  found 
to  be  constant,  and  the  equation  for  the  preservation  of  area  about 
the  vertical  axis  is,  by  a  slight  reduction, 

i^f'  sin^  I  -\-n  cos  ^  z=  B, 

in  which  B  is  an  arbitrary  constant.  The  principle  of  power  gives, 
by  reduction,  the  equation 

2  2 

1/^'  sin^  ^  -j-  ^'  =z  II -\-  «  sin  ^ , 

in  which  II  is  an  arbitrary  constant,  and  «  is  a  constant  which 
depends  upon  the  form  of  the  solid  and  the  radius  of  the  confined 
circumference. 

785.  The  combination  of  (4529)  and  (452i4)  gives 

sin^  ^  ^''  =  {II -\-  a  sin  $)  sin^  ^  —  (B  —  n  cos  If, 

from  which  it  is  obvious  that,  in  the  general  case,  sin  ^  cannot 
vanish,  that  is,  the  hody-cixis  cannot  heeome  vertical. 

786.  When  B  vanishes,  and  H  is  greater  than  a,  we  have 
the  ordinary  case  of  the  devil  on  two  sticks,  and,  in  this  case,  there 
are  three  real  values  of  sin  l,  for  which  the  second  member  of 
( 45220)  vanishes.  Two  of  these  values  of  sin  ^  are  contained  be- 
tween positive  and  negative  unity,  and  one  of  them  is  positive, 
while  the  other  is  negative  ;  they  give  the  limit  of  the  motion  of 
the  axis,  and  correspond  respectively  to  the  cases  in  which  the 
centre  of  gravity  is  below  or  above  the  point  of  dispersion,  which 
latter  is  of  course  the  actual  case  of  the  toy.     In  either  case,  the  end 


—  453  — 

of  the  hody-axis  describes  a  curve  ivMch  is  similar  in  form  io  the  figure  8, 
and  the  apparent  ivant  of  rotation  about  the  vertical  axis  arises  from  the 
repeated  change  in  the  direction  of  rotation  which  occurs  at  each  successive 
return  of  the  bodg-axis  to  the  horizontal  position. 

787.  When  B  vanishes,  and  H  is  greater  than  «,  but   satis- 
fies the  inequality, 

H>^[knaJ  —  n'', 

the  three  values  of  sin^,  for  which  the  second  member  of  (45220) 
vanishes,  are  all  contained  between  positive  and  negative  unity. 
The  positive  value  is  the  upper  limit  of  the  inferior  position  of  the 
centre  of  gravity,  as  in  the  preceding  case,  and  as  it  would  be  if 
the  inequality  of  this  section  were  not  satisfied,  so  that  both  the 
negative  values  were  to  become  imaginary  or  equal.  But  the  two 
negative  values  are  the  limits  of  motion,  when  the  centre  of  grav- 
ity is  higher  than  the  point  of  suspension,  and  in  this  case  tlic  bodg- 
axis  describes  a  ivaving  curve,  and  continues  to  rotate  in  one  and  the  same 
direction  about  the  vertical  axis,  ivithout  ever  becoming  horizontal,  ivhich 
phenomenon  usually  occurs  in  the  devil  on  tivo  sticks,  at  the  beginning  of 
the  game,  and  before  it  has  attained  a  sufficiently  rapid  rotation  to  assume 
a  horizontal  position. 

When  H  satisfies  the  equation 

H=^{^na:f  —  if, 

the  two  negative  limits  of  sin  'S,  are  equal,  and  correspond  to  a 
gyration  of  the  body-axis  about  the  vertical  axis  in  a  right  cone. 
The  motion  which  corresponds  to  a  positive  limit  of  sin  I  in  this 
case  can  be  expressed  by  means  of  elliptic  integrals. 

788.  Whenever  //  satisfies  the  inequality 


—  454  — 

the  body-axis  may  become  horizontal  with  the  centre  of  gravity 
above  the  point  of  suspension,  and  in  this  position  its  gyration 
is  positive  or  negative  in  conformity  with  the  sign  of  B.  If,  more- 
over, n  is  greater  than  B,  the  vibration  of  the  body-axis  from  the 
horizontal  position  extends  so  fir  as  to  reverse  the  direction  of  the 
gyration  about  the  vertical  axis ;  but  if  n  is  less  than  B ,  the  direc- 
tion of  this  gyration  remains  unchanged. 
When  H  satisfies  the  inequalitj^^ 

H<B''-\-a, 

the  body-axis  cannot  become  horizontal  with  the  centre  of  gravity 
above  the  point  of  suspension. 

789.  The  case  of 

B^zhn, 

constitutes  an  exception  to  the  conclusion  of  §  785,  and  it  is 
obvious  that  in  this  case  the  body-axis  may,  and  generally  will, 
become  vertical. 

790.  The  case  of  a  hoop  rolUng  upon  a  horkontal  plane,  is  in- 
cluded in  that  of  any  rolling  solid  of  revolution,  but  which  is  so 
formed  that  a  circumference  of  the  section  of  §  784  is  restricted  to 
roll  upon  a  horizontal  plane.  The  rolling  condition  is  geomet- 
rically satisfied  by  the  restriction  that  the  point  of  contact  with 
the  plane  is  stationary  during  the  instant  of  contact.  If  the  nota- 
tion of  the  preceding  sections  is  retained,  and  if  I  is  the  radius  of  the 
rolling  circumference,  the  velocities  of  the  centre  of  gravity  in  the 
directions  of  the  body-axis  and  of  a  horizontal  perpendicular  to  the 
body-axis  are,  respectivel}'^, 

/r  and/^. 


—  455  — 

The  equation  of  the  power  of  the  system  multiplied  by  2  I^^  and 
divided  by  m  becomes,  then, 


V^'  sin^  ^  +  ^  r  +  ^;  =  H-\-  a  sin  ^  +  i?  n% 
in  which 

n  is  the  initial  velocity  of  rotation  about  the  body  axis,  and  H  is 
arbitrary. 

The  application  of  Lagrange's  canonical  forms  to  the  preceding 
expression  of  the  power  gives  the  equations 

D^  {w  sin- 1  -{-Bv^  cos  ^)  =  0. 
and  by  integration  and  reduction 

Y  sin-  ^  -\-  B  n  cos  ^  =^  C, 
A  sin^  ^  ^' =  (^+  «  sin  ^ )  sin-  ^  —  (  C —  B  n  cos  ^f ; 

and  it  is  obvious  that  these  expressions  coincide  in  form  with  these 
which  were  obtained  in  the  investigation  of  the  devil  on  two  sticks, 
so  that  the  various  inferences  made  in  that  proltlem  are  applicable  to  the 
motion  of  the  hoop.  The  analysis  of  the  present  problem  is  identical 
with  that  wdiich  was  adopted  by  Xulty. 

791.  When  the  hoop  is  gyrating  with  its  plane  in  a  position 
which  is  nearly  horizontal,  the  cube  and  higher  powers  of  sin  \  may 
be  neglected,  in  w^hich  case  the  equation  (4552i)  gives  the  integral 


—  456  — 

in  which  it  is  sufficient  to  observe  that  ^q  is  the  initial  value  of  ^ 
and  1)  is  constant,  80  that  the  hoop  constantly  tends,  hy  its  inertia,  to  rise 
from  this  position,  tvhich,  combined  with  the  irregular  action  of  friction,  ac- 
counts for  the  peculiar  forms  of  gy ration,  tvhich  frequently  accompany 
the  fall  of  the  hoop.  ■ 


ROTARY    PROGRESSION,    NUTATION,    AND    VARIATION. 

792,  The  positions  of  the  axis  of  rotation  and  of  maximum 
rotation-area  may  be  referred  to  a  fixed  axis,  and  the  change  of 
inclination  to  this  fixed  axis  may  be  called  nutation,  while  the  gyra- 
tion about  it  is  called  p?^og?rssion  ; .  and  the  change  in  the  magnitude 
of  the  rotation,  or  of  the  maximum  rotation-area  may  be  called 
variation. 

793.  It  is  obvious  from  the  simple  principles  of  the  computa- 
tion of  rotation-areas,  that  an  accelerative  force  which  tends  to  give 
a  rotation-area  about  an  axis  perpendicular  to  the  axis  of  maximum 
rotation-area,  does  not  cause  a  variation  of  the  rotation-area,  but 
only  a  motion  of  the  axis  so  as  to  incline  it  in  the  direction  of  the 
accelerative  axis.  Hence  if  the  accelerative  axis  is  perpendicular  to  the 
fixed  axis  as  loell  as  to  the  atis  of  maximum  rotation-area,  progression 
is  produced  ;  if  it  is  in  the  common  plane  of  the  fixed  axis  and  axis  of 
maximum  rotation-area,  tvhile  it  is  perpendicular  to  the  latter  axis,  nutation 
is  produced ;  if  it  is  in  the  direction  of  the  axis  of  maximum  7^otation- 
area,  variation  is  produced. 

The  three  directions  of  the  accelerative  axis,  which  correspond 
to  the  respective  production  of  progression,  nutation,  and  variation 
are  mutually  rectangular ;  so  that  it  is  easy  to  determine  the  rela- 
tive tendency  of  a  given  force  to  these  different  modes  of  action. 
This  neat  analysis  is  derived  from  Poinsot. 


—  457  — 

794.  If  the  accelerative  axis  is  constantly  perpendicular  to 
the  fixed  axis,  and  also  to  the  axis  of  maximum  rotation-area,  the 
motion  will  be  wholly  that  of  progression,  of  which  mode  of  action 
a  fixed  iy^e  is  presented  in  the  precession  of  the  equinoxes,  the 
discussion  of  which  problem  must  be  reserved  •  for  the  Celestial 
Mechaxics.  If  the  accelerative  axis  is  constantly  in  the  plane  of 
the  fixed  axis  and  of  the  axis  of  maximum  rotation-area,  while  it  is 
perpendicular  to  the  latter  axis,  the  motion  is  exclusively  that  of 
nutation,  and  this  form  of  action  is  well  exhibited  in  the  friction  at 
the  point  of  the  top  as  it  rolls  upon  the  horizontal  plane. 


ROLLING    AXD    SLIDING    MOTION. 

795.  A  special  example  of  the  case  of  rollmg  motion  has 
been  considered  in  the  hoop,  and  the  mode  of  analysis  which  was  there 
adopted  can  he  applied  to  the  general  investigation,  as  it  has  been  done 
by  NuLTY.  Thus,  let  the  axes  of  x,  g,  z  have  the  same  directions 
with  the  principal  axes  of  the  rolling  solid,  let  Xg,gg,  and  Zg  denote 
the  coordinates  of  the  centre  of  gravity  of  the  solid,  and  x.,g.,  and 
^^  those  of  its  point  of  contact  with  the  surface  upon  which  it  rolls. 
The  condition  of  rolling  without  sliding  gives  the  equation 

with  the  similar  equations  for  the  other  axes.     The  expression  of 
the  power  is 


T=km:^. 


:  Y  [j.  +  {!J-!/.r  +  {^-^^  2  {g-g,)  (.-.,)  ^;  K)\ 


from  which  the  equations  of  nutation  can  be  readily  obtained  by 
Lagrajs^ge's  canonical  forms. 

796.     If  the  solid  slides  upon   the  surface,  it  still  remains  in 

58 


—  458  — 

contact  with  the  surface,  so  that  the  point  of  contact  does  not 
move  in  the  direction  of  the  normal  to  the  surface.  If  the  direc- 
tion of  the  normal  is  denoted  by  N,  this  condition  is  expressed  hy  the 
equation 

which  is  given  by  Anderson.  This  is  the  only  condition,  to  ivhich  the 
motion  is  subject,  in  the  case  of  'perfect  sliding  motion. 

797.  ^Vhen  the  sliding  is  accompanied  ivith  friction,  the  friction 
mag  he  regarded  as  a  force  proportional  to  the  pressure  applied  at  the  point 
of  the  solid,  ivhich  is  in  contact  luith  the  surface,  in  a  direction  opposite  to 
that  of  its  motion. 

When  the  velocity  of  rasure  is  destroyed  by  friction,  the  motion 
ceases  to  be  sliding  and  becomes  a  rolling  motion,  in  ivhich  form  it 
continues  as  long  as  the  force  of  friction  exceeds  the  accelerative  force  in 
the  direction  of  friction. 


CHAPTER  XIII. 

MOTION   OF    SYSTEMS. 

798.  The  motion  of  every  system  is  necessarily  subject  to  the 
Law  of  Poiver,  expressed  in  §  58,  to  the  laiv  of  the  motion  of  the  centre 
of  gravity  of  §  452,  and  to  the  latv  of  areas  of  §  753.  These  three 
principles  not  only  apply  to  the  whole  system,  but  to  each  portion 
of  it  considered  as  a  system  in  itself 

799.  The  various  forces  which  act  upon  a  system  are  often 
quite  different  in  the  magnitude  of  their  effects,  so  that  they  may 


—  459  — 

be  considered  from  this  point  of  view  as  different  orders  of  force. 
In  a  fir^t  investigation,  all  but  the  forces  of  the  first  order  may  be 
neglected  ;  and  in  subsequent  approximations  the  forces  of  the 
inferior  orders  may  be  successively  introduced,  as  disturUng  forces 
and  their  various  effects  may  be  determined  as  pertiirhations  of  cor- 
responding orders. 

800.  The  separation  of  the  system  into  partial  systems  is 
closely  connected  with  this  subdivision  of  the  forces,  for  it  may 
easily  be  seen  that  the  forces,  which  are  of  chief  importance  in 
the  wdiole  system,  or  some  portion  of  it,  are  least  active  in  other 
portions  of  this  system.  Whenever,  for  instance,  the  parts  of  any 
portion  are  so  isolated  from  the  rest  of  the  system,  that  their 
relative  changes  of  position  are  of  small  influence  out  of  the 
portion,  they  should  be  treated  by  themselves  as  a  partial  system, 
and,  relatively  to  all  the  other  parts,  may  be  considered  as  con- 
densed upon  their  common  centre  of  gravity. 


LAGRANGE  S    METHOD    OF    PERTURBATIONS. 

801.  The  method  of  perturbations  which  originated  with  La- 
grange, and  which  depends  upon  the  variation  of  arUtrary  constants, 
deserves  the  first  consideration  from  its  surpassing  elegance  ;  and 
it  is  the  natural  introduction  to  the  other  modes  of  investigation. 

Suppose,  then,  that  a  complete  system  of  integral  equations 
is  obtained,  when  all  the  forces  but  those  of  the  first  order  are 
neglected,  and  let  one  of  these  equations  involving  a  single  arbi- 
trary constant  be  denoted  by  (1992o).     Let 

il  denote  the  potential  of  tlie  forces  of  the  first  order,  and 
T  that  of  the  forces  of  the  inferior  orders, 


—  460  — 
and  the  equations  of  motion  ( 1662-3)  assume  the  forms 

If  the    constant   member  of  (lOOgo)    is   now    assumed   to    vary,  its 
derivative  is 

.       i>,  «i  =  2„  (DJ,  2),  f)  =  S,  [X,  (DJ,  DJ,)  Df^  f] , 

for  by  (19925) 

0  =  ^,  {DJ,  i>„  //-  DJ\  />,  //) . 

The  condition  that  ^  does  not  involve  w  gives  algebraically, 
and  the  notation 

gives  in  combination  with  (4608) 

in  which  a  may  be  substituted  for  its  equal  /  in  the  second  member. 
802.  The  integrals  of  (4602_4)  obtained  with  the  omission  of 
the  forces  of  the  inferior  orders,  admit  of  arbitrary  variation  of  the 
arbitrary  constants,  so  that  if  such  variations  taken  with  reference 
to  arbitrary  elements  which  may  be  denoted  by  y,  and  X,  the  cor- 
responding variations  of  (4602^)  with  the  omission  of  the  terms  de- 
pendent upon  W  are 


—  461  — 
and  similar  equations  for  X  which  give 

SO  that  if 

Cl^'^  does  not  involve  the  time  explicitly. 

803.  If  X  is  the  element  of  actual  variation  of  the  arbitrary 
constants  when  the  inferior  forces  are  introduced,  which  element 
may  be  expressed  as  the  time  when  it  is  so  connected  with  the  arbi- 
trary constants,  as  not  to  cause  ambiguity,  the  variations  of  the 
equations  (4602_^),  give 

D.  t^  =  D,  '^, 

so  that  D^  ^  does  not  involve  the  time  explicitly.  When  x  and  X 
are  changed  to  «j  and  a,,,  it  is  sufficient  to  retain  i  and  k  in  the 
notation   Cj,'^,  so  that  it  is  apparent  from  (461i8)  that 

By  elimination  from  the  equations  represented  in  the  preced- 
ing form,  the  value  of  Dt  a^  can  be  obtained  identical  with  that  of 
(46O20),  so  that  it  is  evident  that  B^^  does  not  contain  the  time  ex- 
plicitly.    It  is  also  apparent  that 

except  when 


—  462  — 
in  which  case 

804.  The  independence  of  B^l^  of  the  time  in  an  explicit  form, 
renders  it  possible  to  compute  its  value  for  any  instant,  and  the 
value  thus  obtained  is  universally  true.  Thus  in  the  especial  case 
in  which  the  arbitrary  constants  are  the  initial  values  of  ?],  w,  etc., 
the  values  of  i>i'^,  computed  for  the  initial  instant,  are  easily  seen 
to  vanish  when  the  k  and  i  refer  to  different  points  of  the  system ; 
but  when  Jc  and  i  refer  to  the  7]o  ^i^cl  ojq  of  the  same  point,  the  value 
of  B^l^  is  positive  or  negative  unity,  so  that 

In  the  case  of  rectangular  coordinates  these  equations  become, 
for  either  axis, 

805.  The  especial  variation  of  the  constant  II  may  be  de- 
rived from  the  equation  (ITl;)  which  gives 

provided  that  t^  is  intended  to  express  the  t  which  is  involved  in 
any  of  the  quantities  denoted  by  i].  This  development  of  the 
variation  of  the  arbitrary  constants  is  taken  from  Lagrange. 


LAPLACE  S  METHOD  OF  PERTURBATION. 

806.     The  values   of  to,  ?;,  etc.,  can   be  substituted  from  the 
first  integrals  directly  in  the  first  form  of  the  second  member  of 


—  463  — 

(46O7),  and  the  integral  values  of  a^  which  are  then  obtained  can 
be  introduced  into  w ,  -^^ ,  etc.,  as  a  second  approximation  to  their 
values.  Tliis  mode  of  analysis  is  especially  useful  when  the  equa- 
tions of  the  first  form  are  linear  with  reference  to  -Jj ,  to ,  and  their 
derivatives.  For  in  this  case  it  is  apparent  that  the  functions  de- 
noted by  /  are  linear  with  reference  to  t],  co,  which  may  be  demon- 
strated in  the  following  manner.  Let  1;,,  w^,  etc.,  be  special  values 
of  1]^  to,  of  which  there  must  be  as  many  independent  values  as 
there  are  equations  expressed  by  (4602_4).  The  arbitrary  constants 
«j-  may  then  be  such  that  the  complete  values  of  i]  and  w  are 

(^  —  ^i  («i  w,) ; 

whence  the  values  of  ojj  assume,  by  elimination,  the  linear  form  in 
reference  to  i],  co,  "etc.  The  values  of  D^^fi,  are  then  functions 
of  t,  and  do  not  involve  rj,  a),  etc.  If  i5^  ^  represent  forces,  which 
are  also  functions  of  the  time,  the  integrals  of  (460;)  can  be  com- 
pletely obtained.  By  the  substitution  of  these  values  of  «j  thus 
obtained  in  the  expressions  of  t],  the  complete  values  of  i]  are  ob- 
tained, which  often  admit  of  useful  modification,  and  the  success  of 
the  method  depends  upon  the  skill  with  which  this  modification  is 
efiected. 

807.     A  special  case  of  frequent  occurrence  in  the  problems 
of  celestial  mechanics  is  one  in  which 

The  value  of  the  integral  in  this  case  is,  for  a  first  approximation, 

tj  =za  cos  at  -\-a-i^  sin  a  i , 


—  464 
whence 


o)  :^  —  a  a  sin  at  A^  aa^  cos  a  t, 
a  =  9^  cos  at sin  a t  =■/, 

czj  =  -J]  sin  « ^  -f-  -  cos  at^=.fi. 


The   values   of  the    constants  obtained  by  integration  of  (4G02_4), 
are  increased  to 

and 

(^i  +  -^ft{D^'^GO^at; 

so  that  the  complete  value  of  i^  is 

9^  =  05  cos^^-j-ojisina/ ^ —  /  [D^  'Fsin«/)-| /  [D^^  Wcosat). 

808.  The  disturbed  motion  of  the  ordinary  projectile  ex- 
hibits an  easy  example  of  change  of  form.  In  this  case,  by  the 
introduction  of  rectangular  coordinates  in  which  the  axis  of  a;  is 
horizontal,  and  that  of  ?/  vertical,  the  equations  are 

whence 

:c^at  +  a,  +  tf,D^W  —  f,{tD,W) 

=.at  +  a,+f?D,W, 
y  =  -hgi'+a,t-^a.,  +  tf,D,W-f,{tD,W) 


—  465  — 


HANSEN  S     METHOD     OF    PERTURBATIONS. 


809.  If  Vf  denotes  any  function  of  the  time  and  of  the  arbi- 
trary constants  in  the  undisturbed  orbit,  its  value  in  the  disturbed 
orbit  may  be  obtained,  from  the  integration  of  the  equation 

by  the  substitution  of  t  for  t  after  the  integration  is  performed. 
In  the  performance  of  the  integration,  the  arbitrary  constants  are 
to  be  regarded  as  variable,  and  the  value  of  F^  in  the  undisturbed 
orbit  is  to  be  taken  for  the  initial  value  of  V^.  This  introdiicUon 
of  T  for  t  constitutes  the  first  frinciple  of  H^vnsen's  method  of  'pcrtur- 
lotions. 

810.  The  application  of  this  method  to  the  example  of  §  808, 
gives,  for  the  values  of  x  and  y 

t 
x.=.at  +  a,-\-f^[{T  —  t}D^T], 

0 

t 

811.  In  the  example  of  §  807,  the  value  of  1]  given  by  this 
method  is 

i]=:a  COS  at  -{-  ai  sin  at /   I  sin  a  (t  —  t)  D   W^  I , 

0 

in  which  the  form  of  notation  is  slightly  modified  so  that  no  subse- 
quent change  of  t  to  z5  is  necessary.  A  case,  which  often  occurs  in 
connection  with  this  example,  is  worthy  of  notice  ;   it  is  when 

D^  W  =ih  cos  {m  t  -\-&), 
59 


—  466  — 

in  which  case  the  value  of  i]  is 

1]  =  a  cos  at  -\-  (Xi  sin  at  -{-  —^ .^  cos  [m  t-\-Ey 

In  the  special  case  of 

m  =  a, 
this  value  of  rj  becomes 

7j  z=.  a  cos  at  -\-  a-^^  sin  at  -\-  —  t  h  sin  (at  -\-  t). 

812.     If  the  function  V  increases  with  the  time  from  negative 
to  positive  infinity,  so  that  for  all  values  of  t 

there  is  an  instant  which  may  be  denoted  by  ^,  for  which  the  un- 
disturbed value  of  V  coincides  with  its  disturbed  value  for  the  in- 
stant denoted  by  t.  The  corresponding  value  of  s-^  is  a  function  of 
both  t  and  r,  which  may  be  introduced  into  V^  instead  of  t,  but 
after  this  substitution  all  the  changes  in  the  value  of  V^  must  arise 
from  those  of  ^^,  so  that 

and  the  difterential  equation  for  the  determination  of  z^  is 


r  '  T 


In  the  integration  of  this  equation,  t  must  be  taken  as  the  initial 
value  of  ;^^,  whence,  for  the  first  approximation, 

T       T 


—  467  — 

After  the  integration  is  performed,  the  value  of  0  is  derived  from 
that  of  z_  by  changing  %  to  t. 

813.  The  disturbed  value  of  any  other  function,  U  may  be 
partially  obtained  by  the  substitution  of  ^  for  ^,  and,  since 

the  residual  portion  is  obtained  from  the  equation 

T  T  T 

by  changing  t  for  t  after  the  integration  is  performed,  and  complet- 
ing the  integration,  so  that  U^  may  be  the  value  of  U^^  when  t 
vanishes. 

This  introduction  of  the  didurhed  time,  ivhich  is  denoted  ly  z,  con- 
stitutes the  second  principle  of  Hansen's  method  of  jjertiirhations,  and 
upon  the  skilful  use  of  the  two  principles  thus  developed,  com- 
bined with  an  appropriate  choice  of  coordinates,  depends  the  suc- 
cess of  this  highly  ingenious  and  original  method. 

814.  It  is  obvious  that,  in  the  first  approximation, 

so  that  the  last  term  of  (467n)  disappears  for  this  approximation. 

815.  If  V  is  such  a  function  that  it  can  be  expressed  in  terms 
of  1^,  etc.,  without  involving  t.»,  etc.,  or  t,  it  follows  from  §801, 
that  the  second  member  of  (4658)  vanishes,  when  t  is  changed  to 
t,  so  that  this  must  also  be  the  case  with  the  second  member  of  the 
equation  derived  from  (46637), 


D 


•^=^m^.4 


—  468  — 

The  value  of  the  first  member  of  this  equation  can  therefore  be 
obtained  by  the  integration  of  the  equation 

provided  that  the  integration  is  completed  in  conformity  with  the 
previous  condition. 

816.  If  one  of  the  arbitrary  constants,  which  may  be  denoted 
by  fi  is  so  involved  in  V  that 

in  which  K  does  not  involve  the  time,  or  if  the  form  of  V  is 

the  corresponding  term  of  the  second  member  of  (4684)  is 

The  corresponding  term  of  the  second  member  of  (467ii),  if  U  has 
the  same  form  with  V —  ^  in  (468ii),  is 

817.  If  one  of  the  arbitrary  constants,  which  may  be  de- 
noted by  Y  is  so  involved  in  U  that  U —  7  may  be  expressed  as  a 
function  of  V  without  explicitly  involving  y  ov  t,  the  corresponding 
term  of  (46 7n)  is  reduced  to 

818.  The  further  development  of  the  methods  of  perturba- 
tions depends  upon  the  peculiarities  of  the  problem  to  which  they 


—  469  — 

are  applied.  But  the  example,  to  which  they  are  most  appropriate, 
is  that  from  which  they  have  derived  their  origin,  the  motions  of 
the  bodies  of  the  solar  system,  so  that  their  ampler  discussion  is  re- 
served for  the  Celestial  Mechanics. 


SMALL    OSCILLATIONS. 

819.  When  the  motion  of  a  system  is  restricted  to  small 
oscillations  about  a  position  of  equilibrium,  the  quantities  i],  etc., 
may  be  supposed  to  be  so  small  that  the  terms  of  T  and  i2,  which 
are  of  more  than  two  dimensions  in  reference  to  these  quantities 
and  their  derivatives,  may  be  neglected. 

The-  value  of  T  may,  then,  by  (IGSg),  be  expressed  in  the  form 

in  which  the  quantities  denoted  by  T^'\  are  constant. 

If  the  values  of  i],  etc.,  are  supposed  to  vanish  for  the  position 
of  equilibrium,  the  derivative  of  S2  with  reference  to  either  of 
these  variables  vanishes  for  the  same  position,  so  that  X2  must  have 
the  form 

in  which  the  quantities,  denoted  by  i2^'^,  are  constant. 

The  equations  of  motion,  derived  from  Lagrange's  canonical 
forms,  are,  therefore,  represented  by 

that  is,  the?/  constitute  a  system  of  linear  differential  equations  tvith  constant 
coefficients. 

820.  It  follows  from  the  linear  forms  of  these  equations,  that 


—  470  — 

the  various  systems  of  values  by  which  they  are  satisfied,  can  be 
combined  by  addition  into  a  new  system.  This  is  the  mathematical 
expression  of  the  important  physical  law  of  the  possibility  of  'the 
siipefyosition  of  small  oscillations. 

821.  With  the  notation 

the  equation  (46928)  assumes  the  form 

If,  then,  there  are  m  of  the  quantities  i] ,  etc.,  if  —  iv'  is  one  of  the 
values  of  D'^^  which  satisfies  the  equation,  expressed  in  the  notation 
of  determinants, 

any  system  of  values  of  tJj  is  expressed  by  the  equation 

'r]i=zEiSm{nt-\-E,,), 

in  which  e,^  is  an  arbitrary  constant,  and  the  constants  Ei  have  a 
common  arbitrary  factor.  The  mutual  ratios  of  the  quantities  E^ 
are  determined  from  the  equations  derived  from  (470io)  by  the 
substitution  of  — n^  for  D'^,  and  E  for  i].  Hence,  by  §340,  E^  is 
determined  in  the  form 

E  ^=E  B^'^^^  , 

in  which  E,,  is  an  arbitrary  constant. 

822.  By  the  combination  of  all  the  values  of  n,  the  complete 
value  of  i].i  is 

7y,  =  ^„  [E,  E^  %,  sin  {n  t  +  £„)]  ; 

but  it  is  evident  that  only  those  values  of  n  should  be  retained 
for   which    the   values   n^   given  by  (470i4)  are  real,  positive,  and 


—  471  — 

unequal.  For  all  other  values  of  ?z^,  the  time  i  would  be  intro- 
duced into  the  value  of  i/^  in  such  a  way  that  it  w^ould  indefi- 
nitely increase.  It  is  plain,  therefore,  that  the  only  values  of  n, 
which  can  be  retained  in  (47028),  are  those  w^hich  correspond  to 
elements  of  stability,  so  that  if  the  elements  i]  are  selected  with 
a  due  regard  to  the  conditions  of  equilibrium,  those  which  corre- 
spond to  the  unstable  equilibrium  will  disappear  of  themselves 
with  the  rejection  of  the  corresponding  values  of  n. 

When  the  j^osUion  of  equilibrium  is  stable  tvith  reference  to  all  of  its 
elements,  all  the  m  values  of  n^  are  real,  ^positive,  and  unequal. 

823.     The  forms  of  T  and  i2  of  §  819,  lead,  by  inspection,  to 
the  equations 

T]^  =  Tl'\ 

and  the  equation  (46928)  gives,  for  each  value  of  n, 

if  n  written  as  an  accent  indicates  a  special  value  of  w,  to  which  the 
functional  form  is  ajDplicable.     If  ^„  is  determined  by  the  notation 

and  if  the  equations,  represented  by  (46928),  ^^^  added  together 
after  being  multiplied  by  F^"\  the  sum  is 

If,  moreover,  T^  denotes  the  value  of  T  when  rji  is  changed  to  IJ}"-\ 
the  value  of  b«,  given  by  integration,  is 

'^,,  =  T„  sin  {nt-\-t,,). 
The  elements  ^  thus  obtained,  correspond  to  the  independent  ele- 


—  472  — 

ments  of  stability  which  affect  the  position  of  equilibrium,  and 
embody  the  true  analysis  of  the  various  forms  of  oscillation  of 
which  the  system  is  susceptible.  When  the  different  values  of  n 
have  a  common  divisor,  the  oscillation  is  evidently  periodic. 

This  investigation  of  the  theory  of  small  oscillation  coincides, 
in  substance,  with  that  of  Lageange. 

824.  The  importance  and  variety  of  the  forms,  in  which 
oscillation  and  vibration  are  physically  exhibited,  give  peculiar 
interest  to  the  mechanical  discussion  of  this  subject.  But  the  mode 
of  analysis  is  so  dependent  upon  the  form  of  the  phenomena,  that 
the  special  researches  are  reserved  for  the  chapters  to  which  they 
are  appropriate. 


A    SYSTEM    MOVING    IN    A    RESISTING    MEDIUM. 

825.  When  a  system  moves  in  a  resisting  medium,  the  law 
of  resistance  may  be  regarded  as  dependent  upon  the  velocity,  so 
as  to  be  the  same  for  all  the  bodies,  but  it  may  vary  by  a  constant 
factor  from  one  body  to  the  other.  If  this  constant  factor  for  the 
mass  lUi  is  denoted  by  /",,  and  if  V^  is  the  function  of  the  velocity 
v.i,  the  resistance  to  the  mass  nii  moving  with  the  velocity  v^  is  /li  F^. 
If,  then,  rectangular  coordinates  are  adopted,  the  equations  of 
motion  assume  the  form 

The  corresponding  form  of  the   equation  for  the  determination  of 
the  Jacobian  multiplier  is,  by  §§  402  and  451, 


A  log  ^J.  =  2,  [k,  ^.  n^^  M\ . 


—  473  — 
This  equation  becomes,  when  the  motion  is  unrestricted  in  space, 

D,  log  ^fo  ^  2.^  [A",  (2  ^  +  A,  V)\  ; 
when  the  motion  is  in  a  plane, 

A  log  ^lo  =  ^.  [/",  (|  +  A,  f;)]  ; 

when  the  motion  is  in  a  straight  line, 

Al0g^fc=:^,(/^A,^)- 

826.  It  is  evident  from  the  linear  form  of  these  equations, 
that  the  multiplier  can  he  separated  into  factors,  each  of  ivhich  shall  inde- 
pendently correspond  to  a  term  of  Vj . 

827.  When  the  resistance  is  constant,  and  the  motion  in  a 
straight  line  ;  or  when  the  resistance  is  inversely  proportional  to 
the  velocity,  and  the  motion  is  in  a  plane  ;  or  when  the  resistance 
is  inversely  proportional  to  the  square  of  the  velocity,  and  the 
motion  is  unrestricted  in  space,  the  multiplier  becomes  unit?/.  In 
either  case  of  motion,  a  term  of  the  corresponding  form  may  be 
added  to  the  resistance  without  affecting  the  multiplier. 

828.  When  the  resistance  is  proportional  to  the  velocity,  the  value  of 
the  multiplier  in  the  case  of  unrestricted  motion  is 

in  the  case  of  motion  in  a  plane  it  is 

and  in  the  case  of  the  straight  line  it  is 

^^=Lc^^iK. 
GO 


—  474  — 

All  these  results,  with  regard  to  the  multiplier,  are  derived 
from  Jacobi. 

829.  When  the  resistance  is  proportional  to  the  square  of  the  velocity, 
the  value  of  the  multiplier  for  motion,  which  is  unrestricted  in 
space,  is 

for  motion  in  a  plane,  it  is 

and  for  motion  in  a  straight  line,  it  is 

830.  The  sum  of  the  equations  (47226),  multiplied  by  nii  x'i,  is 

When  Vi  has  the  form 

the  integral  of  this  equation  is  * 

T—il=  —  :E,\_hm.M^a,t)-\. 

831.  When  there  are  no  external  forces  acting  upon  the  sys- 
tem, the  sum  of  the  equations  (47226)  for  each  axis  multiplied  by  W2j, 
if  x^  refers  to  the  centre  of  gravity,  is 


^,m,D\x,^-:E,i^n,hV:'-^. 


If  the  resistance  is  proportional  to  the  velocity,  the  integral  of  this 
equation  is 

^i  nii  {D,  Xg  —  A)^=z  —  :Si  {nii  ki  x^) , 


—  475  — 

in  which  A  is  an  arbitrary  constant.     If  ki  has  the  same  value  for  all 
the  bodies,  the  complete  integral  is 


kt 


in  which  B  is  an  arbitrary  constant. 

832.  The  introduction  of  polar  coordinates,  and  the  substitu- 
tion of  xl^'^  for  the  product  of  the  area  described  by  lUi  about  the  axis 
of  z,  multiplied  by  the  mass  w?,,  give  for  the  corresponding  equa- 
tions of  motion 

I^,Af  =  DQ,nn—h^^D,Af. 
When  there  are  no  external  forces,  the  sum  of  these  equations  is 

When  the  resistance  is  proportional  to  the  velocity,  the  integral  of 
this  equation  is 

D,^,Af=C—:E,{hA'^), 

in  which  C  is  an  arbitrary  constant,  which  vanishes  if  the  area  van- 
ishes with  the  time.  If  ki  has  the  same  value  for  all  the  bodies 
the  next  integral  is 

^,Af  =  B{l—e-'^). 

So  that  the  rotatioiKtrea  instead  of  being  proportional  to  the  time  is  pro- 
poHional  to 

l  —  c-^\ 

hit  the  position  of  the  axis  of  maximum  rotation-area  is  not  affected  hy 
this  uniform  mode  of  resistance,  which  proposition  is  from  Jacobi. 


476 


THE     CONCLUSION. 


833.  In  the  beginning,  the  creating  spirit  embodied,  in  the 
material  universe,  those  laws  and  forms  of  motion,  which  were  best 
adapted  to  the  instruction  and  development  of  the  created  intellect. 
The  relations  of  the  physical  world  to  man  as  developed  in  space 
and  time,  as  ordered  in  proximate  simplicity  and  remote  complica- 
tion, in  the  immediate  supply  of  bodily  wants  by  the  mechanic  arts, 
and  the  infinite  promise  of  spiritual  enjoyment  by  the  contempla- 
tion and  study  of  unlimited  change  and  variety  of  phenomena, 
are  admirably  adapted  to  stimulate  and  encourage  the  action  and 
growth  of  the  mind.  True  philosophy  begins  with  the  actual,  but 
may  not  remain  there ;  it  yields  sympathetically  to  the  projectile 
force  of  nature,  and  earnestly  forces  its  path  into  the  possible,  and 
even  into  speculations  upon  the  impossible.  But  whenever  the 
initial  impetus  is  exhausted,  the  philosopher  may  not  be  content 
to  remain  stationary,  or  merely  to  turn  upon  his  axis.  He,  then, 
descends  to  the  world  of  sensible  phenomena  for  new  instruction 
and  a  stronger  impulse.  Let  such  be  our  method.  In  the  present 
volume  the  attempt  has  been  made  to  concentrate  the  more  im- 
portant and  abtruser  speculations  of  analytic  mechanics  clothed  in 
the  most  recent  forms  of  analysis,  and  to  make  a  few  additions, 
which  may  not  be  rejected  as  unworthy  of  their  position.  Much, 
undoubtedly,  remains  imperfect  and  unfinished,  for  it  cannot  be 
otherwise  in  a  science  which  is  susceptible  of  infinite  imj)rove- 
ment ;  and  much  must  soon  become  antiquated  and  obsolete  as 
the  science  advances,  and  especially  when  we  shall  have  received 
the  full  benefit  of  the  remarkable  machinery  of  Hamilton's  Quater- 
nions. But  it  is  time  to  return  to  nature,  and  learn  from  her  actual 
solutions  the  recondite  analysis  of  the  more  obscure    problems  of 


—  477  — 

celestial  and  physical  mechanics.  In  these  researches  there  is  one 
lesson,  which  cannot  escape  the  profound  observer.  Every  portion 
of  the  material  universe  is  pervaded  by  the  same  laws  of  mechanical 
action,  which  are  incorporated  into  the  very  constitution  of  the  hu- 
man mind.  The  solution  of  the  problem  of  this  universal  presence 
of  such  a  spiritual  element  is  obvious  and  necessary.  There  is  one 
God,  and  Science  is  the  knowledge  of  Him. 


APPENDIX. 


NOTE    A. 


ON    THE    FORCE     OF    MOVING    BODIES. 


It  is  remarkable,  that,  notwithstanding  the  convincing  argu- 
ments of  Leibnitz,  the  force  of  moving  bodies  is  ahnost  universally 
introduced  into  systems  of  ancLlijtic  mechanics  as  being  proportional  to 
the  velocity,  instead  of  to  the  square  of  the  velocity.  Some  philos- 
ophers, in  quite  an  unphilosophic  spirit,  have  stigmatized  the  early 
discussions  of  this  subject  as  a  war  of  words,  as  if  the  eminent 
geometers  who  entered  into  it  could  have  been  so  deficient  in  their 
powers  of  logic  and  analysis.  The  great  objection  to  the  propor- 
tionality of  the  force  to  the  velocity  is  derived  from  the  necessity 
which  it  involves  of  resiardins;  force  in  one  direction  as  beino;  the 
negative  of  that  which  is  in  the  opposite  direction.  On  this  ac- 
count, when  a  body  or  system  rotates  without  any  motion  of  transla- 
tion, its  aggregate  force  vanishes,  so  that  such  a  motion  would  seem 
capable  of  being  produced  without  any  expenditure  of  force,  and 
this  statement  has  actually  been  made  in  some  works  upon  astrono- 
my. Leibnitz  proposed  as  test  propositions  the  transfer  of  motion 
from  body  to  body  in  various  foiTQS,  in  all  of  which  he  supposed  the 
whole  force  to  be  transferred  from  one  body  to  another  of  a  dif- 
ferent weight  without  any  external  action.  But  it  is  evident  from 
the  law  of  preservation  of  momentum  that  such  a  transfer  is  im- 
possible, and,  therefore,  this  test  cannot  be  practically  applied.  If, 
however,  in    the    case  of  the    impact   of    an   elastic  body  upon  a 


—  480  — 

heavier  one  at  rest,  the  striking  body  is  held  fast,  as  soon  as  it  comes 
to  instantaneous  rest  by  the  transfer  of  all  its  motion  to  the  other 
body,  the  subsequent  action  of  the  elasticity  must  finally  cause  the 
body  which  is  struck  to  move  forward  with  a  velocity  inversely 
proportional  to  the  square  root  of  its  mass.  The  external  effort 
applied  to  the  system  in  this  case  to  hold  the  body  at  rest,  arises 
from  the  force  with  which  the  elastic  spring  of  the  bodies  is  com- 
pressed, and  is  therefore  an  evidence  of  such  a  compression,  and 
a  proof  that  there  has  been  an  expenditure  of  force  in  its  produc- 
tion, although  the  momentum  of  the  system  is  not  changed  until 
the  body  is  held.  If,  again,  a  splierical  ball  were  to  be  impelled  into 
a  cylindrical  tube  of  the  same  diameter,  which  terminates  in  an- 
other cylinder  of  a  different  diameter,  but  which  containing  a  ball 
that  exactly  fits  it,  and  if  the  included  air  acts  as  a  compressed 
spring,  it  is  easy  to  imagine  such  a  mutual  proportion  of  the  parts 
and  weights  that  the  second  ball  shall  leave  the  cylinder  at  the 
very  instant  when  the  first  ball  arrives  at  a  state  of  rest,  and  when 
the  air  has  returned  to  its  initial  density.  In  this  case  the  whole 
living  force  of  the  first  ball  passes  without  increase  or  diminution 
into  the  second  ball,  and  the  momentum  is  not  preserved.  It  is 
true  that  an  external  force  is  required  to  keep  the  cylinders  in 
place,  but  this  is  a  mere  pressure,  which  is  no  more  entitled  to  be 
regarded  as  an  active  force  than  is  the  centrifugal  force,  or  any  of 
the  modifying  forces  which  are  represented  anal3^tically  by  equa- 
tions of  condition.  Seeing,  then,  that  by  admitting  the  square  of 
the  velocit}^  to  be  the  true  measure  of  the  force  of  a  moving  body, 
the  fiction  of  negative  force  is  wholly  avoided,  and  the  funda- 
mental principles  of  mechanical  problems  are  reduced  to  their 
utmost  simplicity,  there  seems  to  be  sufficient  reason  to  reverse  the 
modern  decisions,  and  return  to  the  higher  philosophy  of  Leibnitz. 


—  481  —     . 
NOTE     B. 

ON     THE     THEORY     OF     ORTHOGRAPHIC     PROJECTIONS. 

For  the  convenience  of  students,  the    theory  of  orthographic 
projections  is  here  condensed  into  a  few  simple  formulse. 
The  projection  of  a  line  a  upon  another  line  h  is 

«j  =  a  cos  J. 

If  many  successive  lines  represented  by  f/,,  are  so  united  that 
each  line  begins  where  the  previous  line  ended,  and  if  the  last  line 
terminates  where  the  first  began,  the  sum  of  the  projections  is 

^,(r^cosy=0. 

If  there  are  four  of  these  lines,  and  if  the  three  first  are  mutually 
rectangular  and  parallel  to  the  axes  of  x,  y ,  and  z,  this  equation 
becomes 

2^  {a^  cos  * )  +  «4  cos  ^^  =  0 . 

But  it  is  evident  that  a^  is  the  projection  of  — a^  upon  the  axis 
of  X,  whence 

a^  =  —  ffiCos^^, 

and  if  the  subjacent  4  is  now  omitted  as  unnecessary,  this  equation 
gives 

cos^  =  X^(cos^cos',), 

of  which  the  equation 

l=z^,COS^«, 

is  a  particular  case. 

These  equations  may  be  applied  to  the  projections  of  plane 
areas,  if  each  area  is  represented  in  a  linear  form  by  the  length  of 
a  line  which  is  drawn  perpendicular  to  it. 

61 


ERRATA. 


Page 

For 

Read 

12, 

axes 

axis. 

154-1, 

)  and  1520 

The  signs  of  the  second  member; 

i  should  be  reversed. 

15^4 

acute 

right. 

268 

I 

X,. 

SOu 

these 

those. 

40i8 

resultant 

resultant  moment. 

40^3 

different  lines 

opposite  directions. 

4I22 

force 

resultant  of  the  forces. 

42^ 

0'  with  1 

•eference  to  0 

0  with  reference  to  0'. 

51, 

X, 

x^. 

5I4 

y 

n- 

55i4 

POINT  UPON  A  DISTANT  MASS 

MASS  UPON  A  DISTANT   POINT. 

57i3 

4 

f- 

5722  i 

and  5728 

1. 

2 

|. 

*59, 

cos-^ 

COS'j^. 

5921  ; 

and  60,2 

four 

eight. 

5923 

and  6O21 

two 

four. 

73,6 

surface 

surfaces. 

882 

h 

5.. 

85c 

B^ 

D^- 

8521 

X 

4 

4. 

86u 

4:7t 

4.71  K. 

8622 

l  and  1 

1  and  2. 

8810 

-^t) 

+  ^D. 

90i8 

and  9O20 

ff,a-l 

H^-\ 

9O27 

(cos  (?«  - 

-1) 

cos  {m  —  1)  »/. 

9I25 

See  note  on  page 

356. 

9824 

89r 

8922. 

99io 

rn 

r". 

IOO7 

independent  of 

dependent  upon. 

101,2 

twice 

+  r 

+  2r. 

*  This  correction  only   applies   to   some   copies. 


484 


ERRATA. 


Page 
1^<],9, 16, 18 

107^9 
lllio 

llll3 
llll3 

llll7,  20 

nii9 

117,9 

11722 
117lB 
11721 

1207 
12020 
1211 

1222, 

1285 

126u 
1278 
127.4 
139,, 

U02« 

140.,5 

141i,;.2(, 

148,, 

14822 
1493 

1496 

14?7 

149j8  cos^^si 

149,, 

150.8 

I5O30     twice 

15O31 

151i 

I5I3 

I5I5 

I5I5 

15U 


For 

o 
O 

(104i9)  to  the  form  (107i6) 

■  k 

72 
42,2 

for  another  jioint  of  the  body  which 
is  near  the  former  point 
dele  /ix=. 

iP 

similar  to 
119s 

7t 

above 

put 

in  order 

place 

tan 


sec"  % 

cote 

takei^'^ 

-|-  sin^  t] 

cos-  01  =  cos^  a 

sec"^  0) 


Read 

6. 

Al,. 

(IO49)  to  the  form  (107i). 

k 

rmr'"--^  • 
24. 

4^22* 

arisinof  from  this  motion. 


like. 
II83. 

about. 

but. 

in  order  that. 

plane. 

cot. 


BcB. 

V 
CB. 

sec  *. 
tan  £ . 
take  i^'Z. 

COS^  7/ . 

cot^  CO  =  cot^  a . 


■  ij  sin^  (jp  cos'-^  d  -\-  sin"-^  //  sin^  d    cos'^  d  —  sin*^ »/  sin^  cp  cos^  d  -\-  sin^  ?/  sin^  d  cos-  (jp. 
sin'^  1]  sin  gj  ""'"^  ■ 


6'^ 
-\-  tan  /3 
r  — 

COS^  ^1  (Jp' 

cot  qi' 


sin'  jy  sin'  qp. 
+  • 

d'—. 

—  tan/3. 

d'—. 

—  cos^/3,  g)'. 

(cotg)'  +  ig)')- 


ERRATA. 

48 

Page 

iior 

JJea<i 

152,, 

«'~^ 

V/cos2j3* 

lo2.^, 



+• 

15231 

¥'- 

9"+- 

156^ 

■prolate 

oblate. 

157i 

oblate 

prolate. 

173,, 

for  the 

for  two. 

173,9 

determinate 

determinant. 

175.3 

i'^71  —  1 

t>«  — 2. 

1759 

ap^ 

4>^ 

178,3 

B(rn) 

^w'). 

180, 

®. 

^.. 

I8O7 

«: 

^r. 

190,, 

B 

g^. 

191, 

/' 

/i. 

191 

The  sections  366  to  369  should  be  limited 

vi<^n —  1. 

bj  the  condition  that 

I9I24 

a®;.... 

(%<%[ ^;^>. 

191,5 

(-)"«•:+. 

(         Jn  +  ^nm^l)6^"    ^ 

192^0 

(_).+. 

' 

(—)-». 

192,3 

m  —  1 

m  —  i. 

192i5 

(-)"+' 

/          y„  „  + (i 4-1)  (,„-!-„  4.  1) 

200i„ 

199, 

199a. 

202,, 

111 

2033 

340 

839. 

215^9 

A, 

i>iCi. 

^1^19,24, 

22O3 

Xi,  X2 .  .  . . 

•^ 

3^15   ^2    •    •    •    •   ^?l- 

222i3 

2OO2 

215^2. 

224i 

formal 

normal. 

226n 

;.— 1 

I. 

2272 

(2IO31),  the 

(2IO31).     The. 

2279 

F 

i^. 

228,, 

A 

^^.. 

228^1 

i>.. 

231^5 

187k, 

1893. 

2333 

216a 

231,,. 

2345,7 

D:c,  ^4)(^) 

i?.,  ^(ai)(»>. 

238,9 

^A.. 

^A. 

239 

»ja/(((D 

Jf. 

486 


ERRATA. 


Page 

247^ 

247^5 
25005 

258ij 

261, 

262,, 

26220 

2632, 

2659 

277.2^ 

279; 

2812X 

28I3, 

282^ 

282i4_  18. 

28O20 

287]o_  ig 

287i2 

289,9 

3176 

328-354 

3289 

33O4 

34822 

364j9 

369-370 

37730 

39323 

4262, 

431i9 

47231 


Q 

uniform 

h 

P. 
P' 


sin  (jp 

^{9  —  (^) 
Cot 
kt 
Ra 

correct 

1 

Sj  71 

B  rachy  stochrone 

is  confined 

3299 

589 

siny 

35935 

Tachystotrope 

level 

sint-« 

g)^ 

.Q2 

2>, 


Read 
^. 

uniform  and  constant  in  direction. 

h. 
h_ 

V" 

?r. 

2a2. 

.+. 

cot. 

cosqp. 

y/[A(^_a)]. 

Tan. 

{kt  —  'la). 

Ra         CPo     •     o 

9         k 

nearly  correct. 

—  a. 

fim. 

3 

K 

A  +  2/.\ 

n. 

Brachistochrone. 

is  not  confined. 

3293. 

588. 

cos  V  . 

3593. 

Tachistotrope. 

given. 

sint-iJ. 

^'. 

^,. 


ALPHABETICAL   IXDEX. 


Abel,  method  of  investigating  the  holo- 

chrone,       .....  356 
Acceleration   of    rotating   cylinder   upon 
which  a  body  moves,  when  it  is 
uniform,      .....  254 
of  a  point  by  a  moving  line,      .        247 
Action  and  Reaction,        .         .         .         .132 
of  moving  bodies,      .         .         .        1G2 
principle  of  least, .         .         .         .167 
Andersox  on  rolling  and  sliding  motion,  458 
Archimedes,  spiral  of,  described  by  ac- 
tion of  central  force,  .         .         .  384 
Area,    constant    area     described    by     a 
point  upon  a  surface  of  revolu- 
tion,     412 

in  the  motion  of  a  free  point,    .        424 
constant,  when  all  the  forces  are 

directed  towards  the  axis,  .  .  425 
of  rotation,  ....  433 
conservations  of,  .  .  .  .  434 
of  rotation  for  a  principal  axis,  436 
of  rotation  when  it  Is  a  maximum 

for  a  solid, 437 

of  rotation  expressed  by  Euler's 
equations   for  the   motion  of  a 
solid,  .         .         .         .         .         .437 

of  rotation,  its  axis  when  it  Is  a 
maximum  for  the  free  motion  of 

a  solid, 439 

of  rotation  described  by  the  gyro- 
scope about  the  vertical  axis,      .  445 
of  rotation  of  gyroscojje  affected 

by  friction,           .         .         .         .450 
of  I'otation  of  the  devil,     .         .        452 
of  rotation   of  the  devil,  when  it 
vanishes, 452 


Paoe 

Areas,  principle  of,  in  a  moving  system,    . 

458 

Astronomical  Journal,  see  Gould. 

Asymptote  of  the  brachistochrone  of  in- 

finite branches,  .         .         .         . 

333 

Attraction  of  an  infinite  lamina,    . 

46 

of  an  infinite  cylinder,  . 

49 

of  a  point  upon  a  distant  mass, 

55 

of  a  spherical  shell,  . 

56 

of  a  Chaslesian  shell,    . 

58 

of  an  ellipsoid,  .... 

69 

of  a  spheroid,         .         .         .         . 

88 

Axis  of  rotation, 

12 

of  rotation,  instantaneous. 

19 

of  gravity,           .... 

50 

of  principal  expansion, 

118 

of  Inertia  principal,   . 

436 

instantaneous   in  a  body    and   in 

space,          .... 

439 

B. 

Bailey  on  the  force  of  resistance  to  the 

motion  of  the  pendulum,     .         .  291 
experiments  on  the  motion  of  pen- 
dulum of  various  forms  in  air,     .  311 
Barnard  on  the  gyroscope,        .        .      447 
Bary trope  discussed,          .         .         .         .370 
Bernoulli,  John,  on  the  synchrone, .       373 
Bernoulli,  James's,  lemnlscate,   .         .  380 
Bertrand  on  the  tautochrone,    .         .       364 
Bessell  on  the  resistance  of  the  pendu- 
lum,    292 

experiments    upon    the    seconds' 

pendulum, .....  298 

BoBiLLiER  catenary  on  cone,       .         .       153 

catenary  on  sphere,       .         .         .157 

Bonds  of  union  of  a  rigid  system, .         .       126 

Bonnet,  theorem  of  combination  offerees,  430 


488 


ALPHABETICAL   INDEX. 


Booth,  elliptic  integrals,         .         .         .147 
BoRDA,  experiments  on  the  seconds'  ^len- 

dulum, 296 

Brachistoclirone,   .         .         .         .         .328 
on  the  surface  of  revolution,  .  334 

Brass  sphere  vibrated  by  Bessell,       .       298 
spheres,   cylinders,   and  bars  vi- 
brated by  Bailey,    .         .         .  311 

C. 

Canonical  forms  of  the  equations  of  mo- 
tion,     163 

Catenary, 134 

on  surface  of  revolution,  .  .  143 
on  riiiht  cone,  ....  144 
on  ellipsoid,  .  .  .         .         .154 

on  equilateral  asymptotic   hypcr- 

boloid, 159 

curious  relation  to  the  motion  on 
an  hyperbola  when  the  central 
force  is  proportional  to  the  dis- 
tance,   385 

CArCHY  on  elasticity,  .         .         .         .124 
solution  of  partial  differential  equa- 
tions,   201 

on  differential  equations,  .         .       214 
Centre  of  gravity,     .         .         .         .         .55 
its  position  with  regard  to  equilib- 
rium of  rotation,          .         .         .131 
[resultant  moment  for,        .         .       133 
motion  of,      .         .         .         .         .262 
of  systems,         ....       458 
of  a  system  in  a  resisting  medium,  474 
Central  force  of  gravitation,     .         .         .43 
in  relation  to  tautochrone,         .       323 
in  relation  to  brachistochrone,      .  330 
for  a  point  moving  upon  a  plane,     3  78 
special  cases  of,  which  admit  of  in- 
tegration,     379 

forms  which  admit  of  general  in- 
tegration,  .         .         .         .         .383 
forms  which  admit  of  integration 

by  elliptic  integrals,    .         .         .389 

third  form  which    can  be  solved 

by  elliptic  integrals,    .         .         .  406 

Centrifugal  force, .....       245 

for  brachistochrone,       .         .         .  329 


Centrifugal  force  of  body  moving  on  sur- 
face, .         .         .         .         .         . 

Characteristic  function  of  motion, 

its  variation,      .... 
Chasles's  shell  and  its  attraction,   . 

and  Gauss's  theorem, 

trajectory  canals,  .         .         .         . 

analogy  of  attraction  and  propa- 
gation of  heat,     .         .         .         . 

definition  of  his  thin  shells, 

potential  of  his  shells,    . 

his  ellipsoidal  shell,   . 

attraction  of  his  ellipsoidal  shell,  . 
Circle  rotating  Avith  a  free  moving  body 
upon  its  circumference, 

upon  which  a  heavy  body  moves, 

rotating  in  a  vertical  position,  with 
heavy  body  moving  along  its  cir- 
cumference,       .         .         .         . 

rotating  with  heavy  body  moving 
on  its  circiunference,  . 

involute,  with  body  moving  along 
it  against  resistance,    . 

involute,  a  case  of  tautochrone, 

a  tautobaryd,         .         .         .         . 

described  in  a  case  of  central 
force, 

horizontal,  when  it  is  in  the  path 
of  a  pendulum,   .... 

great,  when  it  is  nearly  the  path 
of  a  pendulum,  .... 

general  law  of  description, 

section  of  ellipsoid  of  inertia  de- 
scribed by  axis  of  maximum  ro- 
tation area  of  a  solid, . 

path  of  the  gyroscope, 
Clairaut  on  a  case  of  the  tachytrope,    . 

motion  of  a  body  when  the  central 
force  is  inversely  as  the  cube  of 
the  distance,        .... 

Composition  of  forces,    .... 

Conclusion,        ...... 

Condition,  equations  of,         .         .         . 
Cone,  catenary  upon,        .... 

tautochrone  of  heavy  body  upon, 

brachistochrone  of  heavy  body 
upon, ...... 

motion  of  heavy  body  upon. 


377 

162 

166 

58 

61 

63 

64 
65 
61 
70 
76 

251 
255 


259 

264 

274 
325 
372 

379 

419 

421 
432 


442 
446 
365 


388 
40 

476 
24 

144 

322 

341 
413 


ALPHABETICAL   INDEX. 


489 


Conic  section  described  when  the  central 
force  is  proportional  to  the  dis- 
tance,   

described  when  the  central  force 
is  inversely  proportional  to  the 
square  of  the  distance, 
described     when     many    central 
forces  act  proportionally  to  the 
distance,     .         .  -     . 
general  law  of  description, 
Conservation  of  power,     .         .         .         . 
of  motion  of  centre  of  gravity,  . 
of  areas,        ..... 
of  power  in  motions  of  systems, 
Constants,  variation  of  arbitrary. 
Continuity,  solution  of,  in  cases  of  resist- 
ance, .  .         .         .         . 
in  the  potential  of  nature, . 
Coordinates,  peculiar  case  of,    . 
Couple  of  rotations,       .... 

of  forces,       .         .         .         .         . 
Cusps  of  brachistochrone. 
Cycloid  the  tautochrone  of  a  heavy  body, 
the  base  of  a  cylinder  on  which 

lies  a  tautochrone, 
meridian  curve  of   a   surface   of 
revolution  on  which  lies  a  tau- 
tochrone,   .         .         .         .         . 
the   brachistochrone    of  a   heavy 

body, 

a  tachytrope,         .... 
conditions  of  description, 
Cylinder,  attraction  of,      . 

containing  a  catenarj', 

rotating  with  a  body  moving  upon 

a  given  line  of  its  surface,     . 
having  a  heavy  body  upon  its  sur- 
face,       

vibrated  by  Bessell,  . 
vibrated  by  Baily, 
containing    a    tautochrone    upon 
its  surface,     .... 


385 


38G 


425 
432 
163 
242 
434 
458 
459 

273 
32 

425 
18 
40 

332 

318 

321 


323 

332 
365 
432 
49 
143 

253 

254 
298 
311 

319 


D. 

Derivative  multiple, . 
Determinants,  theory  of, 
functional,     . 


196 
172 
183 

62 


Determinants  applied  to  multiple  deriva- 
tives and  Integrals,      .         .         .196 

Devil  on  two  sticks,     .         .         .         .         451 

DuBUAT  on  the  law  of  resistance  of  a 

medium,        ....         292 
experiments     on    the     pendulimi 
against  a  resistance,       .         .         294 

DtJPiN  on  orthogonal  surfaces, .         .         .79 

E. 

Economy  dynamic,  of  nature,  .         .         .168 
Elasticity,    .         .         .         .         .         .         116 

Electricity,  statical,  .         .         .         .         .44 

Ellipse,  spherical,        .         .         .         .         147 

described  by  central  force  which 

is  proportional  to  distance,          .  385 
described  under  the  law  of  gravi- 
tation,          386 

Ellipsoid,  attraction  of,  .         .         .         69 

Chaslesian  shell,  .  .  .  .70 
of  revolution,  attraction  of,  .  87 
of  closest  approximation  to  at- 
traction of  spheroid,  .  .  .103 
of  expansion,  .  .  .  .  118 
of  reciprocal  expansion,  .  .121 
with  catenary  upon  its  surface,  154 
with  brachistochrone  on  its  surface,  344 
defining  surface  of  the  brachisto- 
chrone, .  .  .  .  .347 
of  Inverse  inertia,    .         .         .         435 

of  inertia, 436 

Elliptic  integrals  for  attraction  of  ellipsoids,  83 
for  the  catenary  upon  the  cone,  .  147 
referred  to  spherical  ellipse,  .  149 
for  the  catenary  upon  the  sphere,  157 
for  the  simple  pendulum,  .  .256 
for  tautochrone  on  a  moving  curve,  318 
for  tautochrone  on  a  cycloidal  cyl- 
inder,   321 

for  brachistochrone  with   parallel 

forces, 333 

for  brachistochrone  on  jiaraboloid,  337 
for   brachistochrone    on   inverted 

paraboloid,  .         .         .         .341 

for  brachistochrone  on  cone,  •  343 
for  brachistochrone  on  sphere,  .  346 
for  circular  brachistochrone,   .         354 


490 


ALPHABETICAL   INDEX. 


Elliptic  integrals  for  two  forms  of  central 

force, 380 

for  third  form  of  central  force,  .  406 
for  motion  upon  a  cone,  .  .413 
for  motion  upon  a  paraboloid,  .  416 
for  motion  upon  an  inverted  para- 
boloid, .  .  .  .  .  417 
for  the  time  of  spherical  pendulum,  418 
for  the  azimuth  of  the   spherical 

pendulum,  .         .         .         .423 

for  forms  of  force  directed  towards 

axis, 428 

for  rotation  of  a  free  solid,         .      442 

for  the  gyroscope,  .         .         .  446 

for  the  top,        ....       450 

Epicycloid  a  tautochrone,         .         .         .327 

a  brachistochrone,     .         .         .331 

path   described  under  action    of 

central  force,      .         .         .         .379 

Equation  of  tendency  to  motion,  .         .         lis 

of  motion,  ditferential,  .         .         •     Sjg 

of  equilibrium,  .         .         .         .         Sjjo 

of  orthogonal  cosines,    .         .         .1520 

of  instantaneous  axis  of  rotation,      1 7j5 

of  rotation  for  cylinder,         .         .  23i4 

of  condition  involved  in  those  of 

motion  and  rest,  .         .         .  26i2 

of  condition  referred  to  normal,       27i5 
of  tendency  to  motion  expressed 

by  potential,  ....  3431 
of  resultant,  .  .  .  .  372i 
of  potential  of  gravity,  .  .  459 
Laplace's,  of  potential,  .  .  463 
Laplace's,  modified  by  PoissoN,  492 
of  potential  of  an  infinite  cylinder,  4931 
of  relation  of  potential  to  its  para- 
meter,          554 

of  Gauss,  for  action   normal   to 

surface, eOoi 

of  attraction  of  ellipsoid  in  direc- 
tion of  either  axis,      .         .         .  82o4 
Legendre's,     for    attraction    of 

ellipsoid, 83i2 

of  Legendre  upon  attraction,  .       8610 
of  function  for  expression  of  the 

attraction  of  an  ellipsoid,    .         .  8602 
of  attraction  of  a  homogeneous  ob- 
late ellipsoid  of  revolution,         .  8730 


Equation  of  attraction  of  a  homogeneous 

j^rolate  ellipsoid  of  revolution,    .  88ig 
of  function  developed  in  cosines 

of  multiple  angles,       .         .         .  89i3 
of  elementary  functions  of  Legen- 
dre's functions,  .         .         .  9324 
of  Legendre's  functions  in  spe- 
cial form,    .         .         .         .         .993 
of  theorem   for  development   into 

Legendre's  functions,    .         .  IOI22 
Laplace's     upon     Legendre's 

functions, 102,) 

Laplace's  more  general  form  of 

Legendre's  functions,    .        .  IO220 
of  potential  of  ellipsoid  referred  to 

centre  of  gravity,      .         .         .  lOSjo 
of  Legendre's  second  function,    IO49 
of  external  j^otential  of  spheroid 
with  the  introduction  of  ellipsoid 
of  nearest  attraction,         .         .10724 
for  axes  of  nearest  ellipsoid  of  at- 
traction,     IO83 

of   potential  for  point  near   the 

spheroid,  .         .         .         .         .    IIO3 
Laplace's,   for   spheroid    which 

differs  little  from  a  sphere,  .  1152? 
of  ellipsoid  of  expansion, .  .  1 1 83 
of  surface  of  distorted  expansion,  119i3 
of  total  expansion,  .  .  .  12O27 
of  ellipsoid  of  reciprocal  expan- 
sion,   12I11 

of  equilibrium  of  translation,  .  12  730 
of  funicular,  ....  138i4 
of  catenary,  ....  I3827 
of  extensible  catenary,  .  .  14115 
of  catenaiy  upon  a  surface,  .  1420 
of  pressure   of   catenary  upon   a 

surface, 14  2,0 

of  catenary  upon    a    surface    of 

revolution,  ....  14429 
of  arc  of  spherical  ellipse,  .  .  14920 
of  total  expenditure  of  action, .  1 6  2,1 
of  living  forces,  .  .  .  .  163i4 
Lagrange's  canonical,  of  motion,  164i2 
Hamilton's  changes  of  La- 
grange's canonical  forms,  .  165o7 
for  characteristic  and  principal 
functions,  .         .         .         .  17I20 


ALPHABETICAL  INDEX. 


491 


Equation  of  determinants,  .  .  .  173, 
linear  solved  by  determinants,  .  177 
simultaneous  differential,   related 

to  linear  partial  differential,  .  199 
differential  in  normal  form,  .  210^ 
partial  differential  for  Jacobian 

multiplier,  ....  215i2 
oomuion  differential  for  Jacobian 

multiplier,  .  .  .  .  216, 
of  Jacobian  multiplier  for  equa- 
tions of  motion,  .  .  .  237i(, 
of  translation,  .  .  .  .2422 
of  time  of  describing  a  line,  .  243.,; 
of  centrifugal  force,  .         .       245i8 

of  motion  upon  a  rotating  line,  .  247oo 
of  motion  of  a  heavy  body  upon  a 

moving  line,  .  .  .  .  257^ 
of  gain  of  power  by  motion  of  the 

line  of  support,  .         .         .  25925 

of  motion  of  a  fixed  line  through 

a  resistance,  .  .  .  .  27I3 
of  motion  against  friction,  .  273ir, 
of  fixed  force  for  tautochrone,  .  317,, 
of  tautochrone  for  central  force,  323i5 
of  general  brachistochrone,  .  328i8 

of  brachistochrone  for  fixed  force,  328^7 
of  brachistochrone  for  radius  vec- 
tor and  jierpendicular  from  origin 
in  central  force,        .         .         .  33O15 
of   brachistochrone     for    parallel 

forces, 331,1 

of  brachistochrone    on  surface  of 

revolution  for  central  force,      .  3242s 
of  brachistochrone  of  given  length,  34  7;jo 
of  brachistochrone  of  given  expen- 
diture of  action,        .         .         .  349i(, 
of  the  holochroue  when  the  time  is 
a  given  function  of  the    poten- 
tial,   357i4 

of  tautochrone  from  Lagrange,  36I2 
of  tachytrope,  ....  364i; 
of  tachytrope  for  central  foi'ce  in 

resisting  medium,      .         .         .    3C64 
of  tachistotrope   in  resisting  me- 
dium,          369,5 

ofbarytrope,   ....       370^1 
of  path  of  a  point  upon  a  surface 
with  fixed  forces,       .         .         .377; 


Equation  of  path  of  a  body  when  the  force 

is  central,  ....  37802 

of  path  of  a  body  upon  a  surface 
of  revolution  with  central  force 
directed  toward  the  axis,       .     4129,17 
of  the  spherical  pendulum,       .    41 817.01 
of  force  for  the   description   of  a 

given  curve,      ....  43O20 
of  EuLER  for  rotation  of  a  solid,    437oo 
of  living  force  in  a  rolling  solid, .  45  737 
of  sliding  motion,          .         .         .  4585 
of  variation  of  arbitrary  constants,  46O30 
of  variation  of  initial  values  of  va- 
riables       ....        462ii^i2 
of  Hansen's  method  of  perturba- 
tions,    .         .         .     465^,  46625,  467i(, 
of  small  oscillations,     .         .         .46927 
of  multijilier  in  a  resisting  medium,  4  72,1 
of  power  in  a  resisting  medium,  .  474i5 
of  translation    of  a   resisting  me- 
dium,          474o7 

of  rotation  in  a  resisting  medium,  475ii 


Equilibrium,  equations  of,         .         .         . 

7 

conditions  of,     . 

29 

stable  or  not,         .         .         .         . 

30 

of  translation,    .... 

127 

of  rotation, 

129 

oscillation  about  position  of. 

471 

Euler,  integral, 

91 

note  on  erroneous  notation. 

356 

on  differentia]  equations. 

214 

centrifugal  force  on  the  brachisto- 

chrone,        

329 

on  the  brachistochrone  of  central 

forces, 

330 

on  epicycloidal  brachistochrone. 

331 

error    regarding     the     brachisto- 

chrone,         

353 

compound  brachistochrone. 

354 

compound  tautochrone. 

358 

tachytrope  of  heavy  body. 

364 

tachytrope  for  parallel  forces. 

366 

tachytrope  of  constant  velocity  in 

a  given  direction. 

367 

tachistotrope  of  heavy  body, 

369 

tautobaryd  of  heavy  body,    . 

373 

path   of  body  gravitating  to  two 

centres, 

429 

492 


ALPHABETICAL  INDEX. 


EuLER,  equations  of  rotation  of  solid,  .  437 
rotation  of  solid,    ....  443 

Evolute  of  the  parabola  a  tachytrope,   .       368 

Expansion,  linear,  .  .  .  .  .117 
ellipsoid  of,  .  .  .  .118 
surface  of  distorted,  .  .  .119 
total, 120 

Expenditure  of  action,      .        .         .        .162 

F. 

Fontaine  on  tautochrone,       .         .         .  362 

Force,  its  origin,    .....  1 

measure  of,  .         .         .         .         .2 

of  moving  bodies,       ...  4 

of  nature, 28 

fixed, 28 

expressed  in  form,  .  .  .29 
potential  of,  .  .  .  .  29 
temporarily  fixed,  .         .         .34 

composition  and  resolution  of,  .  35 
moment  of,  .  .  .  .  .38 
couple  of, .  .  .  .  .  40 
in  a  plane,    .         .         .         .         .42 

parallel, 42 

modifying,     .  .         .         .         .124 

internal,  maybe  neglected  in  trans- 
lation and  rotation,     .         .        .131 
equal  and  parallel,  in  equilibrium,  132 
principle  of  living,         .         .  .163 

of  perturbation,  .  .  .  459 
of  moving  bodies,  .         .         .479 

central.  See  Central  Force. 
centrifugal.  See  Centrifugal  Force. 
Form,  expressive  of  force,  .  .  .29 
French,  weights  and  measures  introduced,  293 
Friction  opposing  n'iotion  of  a  body,  .  .270 
changing  sliding  to  rolling  motion,  458 
Functional  determinant,  .  .  .  .183 
Funicular,     .         .         .         .         .         .134 

G. 

Gamma  function,  .  .  .  ,  .91 
note  on, 356 

Gauss  on  action  perpendicular  to  surface,  60 
maxima  and  minima  of  potential 

of  gravitation,    .         .         .         .62 
determinants,    .        .         .         .173 


Gould's  Astronomical  Journal,  on  partial 

multipliers,    .         .  .  .         .231 

on  motion   when  force    emanates 

from  an  axis,  ....  428 

Gravitation,  jiotential  of,  .         .         .         .43 

potential  for  mass,     ...  45 
the    type    of  equal   and  parallel 

forces, 132 

its  level  surfaces,       .         .         .  132 

Gravity.     See  Centre  of  Gravity. 

GuDERMANN  On  Spherical  pendulum,       .  423 

Gyration  of  the  devil,    ....  453 

of  the  hoop, 456 

Gyroscope, 443 

H. 

Hamilton's  characteristic  function,         .  162 
on  Lagrange's  canonical  forms,  164 
modification   of  Lagrange's  ca- 
nonical forms,  .         .         .       165 
principal  function,         .         .         .169 
new  method  of  dynamics,  .        .       171 
quaternions,          .         .         .         .476 
Hansen,  method  of  perturbations,        .       465 
Helix,  rotating  with  body  moving  upon  it,  254 
Holochrone,      ......  354 

Hoop,  motion  of, 451 

Hyperbola,  determining  the  limits  of  mo- 
tion on  a  rotating  circumference,  265 
described  by  central  force,         .       380 
described     by    repulsive     central 

force  proportional  to  distance,     .  385 
described  by  force  of  gravitation,    386 
Hyperboloid  equilateral    asymtotic,    con- 
taining catenary,         .         .         .  159 
defining   limits  of  catenary  upon 

other  surfaces  of  revolution,  .  160 
homofocal  with  ellipsoid,  .  .  77 
containing  brachistochrone,  .         .347 


Inertia  of  matter, 1 

moment  of,         ....       434 

Integral  multiple,  .  .  .  .  .197 
of  diiferential  equations,  .         .       199 

Integrals,  systems  of,  ...  .  203 
elliptic.     See  Elliptic  Integrals. 


ALPHABETICAL   INDEX. 


493 


Integration  of  the  differential  equations  of 

motion, 172 

Involute  of  circle,  described  in  a  resisting 

medium,      .         .         .         .         .274 

a  tautochrone,  ....       325 

Ia^ory  on  corresponding  points,        .         .     70 

Ivory  sphere  \ibrated  by  Bessel,        .       298 


Jacobi  on  Legendre's  functions,  .  .  88 
on  determinants,  .  .  .195 
on  normal   forms    of   differential 

equations,  .  .  .  .  .210 
new  multiplier, .  .  .  .  214 
principle  of  last  multiplier,  .  .  228 
on  the  motion  of  a  body  in  a  resist- 
ing medium,  ....  376 
on  motion  of  a  body  gravitating  to 

two  fixed  points,         .         .         .  429 
on  motion   of  a  system  in  a  re- 
sisting medium,    .         .         .         .474 
Jellett  on  the  tangential  radius  of  curva- 
ture of  the  brachistochrone  on  a 

surface, 347 

on  the  brachistochrone  of  a  heavy 
body  in  a  resisting  medium,        .  353 

K. 

Klixgstierxa's  problem   of  the  tachy- 

trope, 365 


Lagrange,  method  of  mechanical  analysis,      9 
canonical  forms  of  equations  of  mo- 
tion,    165 

on  determinant  of  derivatives,  .       194 
on  differential  equations,       .         .  214 
modification  of  Euleriax  multi- 
plier,   232 

on  the  tautochrone,  .  .  .  359 
familiar  formula  of  the  tautochrone,  361 
on  the  rotation  of  a  solid,  .  .  443 
on  the  motion  of  a  body  gi-avitat- 

iug  to  two  centres,      .         .         .  429 
on  the   method    of  perturbations 
by  the  variation  of  arbitrary  con- 
stants,   459 


Lagrange  on  small  oscillations,  .  .  472 
Lamina,  attraction  of  infinite,  .  .  .46 
Lame',  relation  of  potential  to  its  parameter,  55 
Laplace,  equation   for  the  potential  of 

gravitation,  .         .         .         .46 

equations  modified  by  Poissox,        48 
attraction  of  Newtonian  shells,  .     75 

functions, 88 

theorems  on    Legendre's   func- 
tions,   102 

equation      for     nearly     spherical 

spheroid, 115 

on  the  tautochrone,   .         .         .       360 

on  the  rotation  of  a  solid,       .         .  443 

method  of  perturbations,       .         .  462 

Lead  sphere,  vibrated  by  Newton,      .       293 

Legendre,   attraction    of  Newtonian 

shells, 75 

attraction  of  ellipsoids,       .         .         83 
theorems  on  the  attraction  of  ellip- 
soids,        86 

functions,      .         .         .         .         .88 
special  form  of  functions,   .        .         99 
Leibnitz  on  the  force  of  moving  bodies,    479 
Lemniscate,  described  under  law  of  cen- 
tral force, 380 

Level  surfaces,      .         .        .         .        ,         32 

of  gravity, 132 

a  syntachyd,      .         .         .         .375 
Limits  of  brachistochrone,          .         .         .  348 
of  body  moving  under  central  force,  40  7 
of  heaA  y  body  on  surface  of  revo- 
lution,          413 

Linear  equations  solved  by  determinants,  177 
partial  differential  equations,         .  199 
equations  of  small  oscillations,   .       469 
Logarithmic  spiral  described  by  a  body  on 

a  rotating  straight  line,  .  .251 
described  against  resistance,  .  274 
a  tautochrone,  .  .  .  .325 
a  tachy trope,  .  .  .  .365 
described  under  the  action  of  a 
central  force,      .         .         .         .379 

M. 

Maclaurin's  attraction  of  ellipsolil,        .     75 

Mass  defined, 2 

Matter,  inertia  of,     .         .         .         .         .1 


494 


ALPHABETICAL  INDEX. 


Maupertius,  action  of  a  system, 

162 

principle  of  least  action, 

168 

Maximum  and  minimum  of  potential,    . 

29 

for  equal  and  parallel  forces, 

132 

of  velocity  of  pendulum  in  a  re- 

sisting medium. 

283 

Measures,  French  adopted, 

293 

Medium,  resisting,          .... 

270 

bracliistoclirone  in, 

350 

holochrone  in,  , 

359 

tachytrope  in,        .         .         . 

364 

tachistotrope  in,         .         .         . 

369 

barytrope  in,         .         .         . 

371 

tautobaryd  in,    . 

371 

synchrone  in,        .         .         . 

374 

syutachyd  in,     . 

375 

systems  moving  in, 

472 

Method  of  multipliers,   .... 

25 

Hamilton's,  of  dynamics,  . 

162 

Lagrange's,  of  perturbation,  . 

459 

Laplace's,  of  perturbations. 

462 

Hansen's,  of  perturbations. 

465 

Modifying  forces,       .... 

124 

Moment  of  force, 

38 

resultant,       .... 

39 

of  inertia, 

435 

Motion  necessary  to  phenomena, 

1 

uniform, 

2 

measure  of,  . 

2 

tendency  to,      . 

5 

equation  of,  . 

7 

perpetual,  impossible  in  nature, 

31 

of  translation,        .         . 

241 

of  a  point,          .... 

242 

of  rotation,    .... 

433 

of  a  system,        .... 

458 

IMultiple  derivatives  and  integrals,    . 

196 

Multiplier,  method  of,   .... 

25 

Jacobian,   .... 

214 

principle  of  last. 

228 

for  equations  of  motion, 

236 

for  motion  of  a  point. 

244 

for  motion  in  a  resisting  medium. 

472 

N. 


Nature,  forces  of, 
Newton's  shell. 


Newton's  experiments  on  pendulum,      .  293 

path  described  when  the  central 

force  is  inversely  as  the  cube  of 

the  distance,  .        .         .         .379 

Normal  form  of  differential  equations,  .       210 

Notation  of  reference,       ....      4 

NuLTY  on  the  hoop,      ....       455 

on  rolling  motion,  .         .         .457 

Nutation  of  rotation,      ....      456 

O. 

Orthogonal  surfaces,  .         .        .        .79 

Oscillations  about  position  of  equilibrium,     30 

of  a  body  on  a  fixed  line,      .         .  246 

of  a  bod}'  on  a  uniformly  rotating 
line, 248 

on  a  rotating  circumference,     .       252 

of  the  pendulum,  ....  256 

of  a  heavy  body  on  a  rotating  cir- 
cumference,       ....  266 

of  the  pendulum  when  the  resist- 
ance is  proportional  to  the  veloc- 
ity,       282 

of  the  pendulum  when  the  resist- 
ance is  proportional  to  the  square 
of  the  velocity,   .         .         .       -.  285 

of  the  pendvdum  with  the  medium,  287 

of  the  pendulum  when  opposed  by 
friction, 290 

of  the  pendulum  observed  by 
Newton, 293 

of  the  pendulum  observed  by 
DuBUAT, 295 

of  the  pendulum  observed  by 
BORDA, 296 

of  the  pendulum  observed  by 
Bessel, 298 

of  the  pendulum  observed  by 
Bailey, 311 

small,  theory  of,         .         .         .469 


Paper  sphere  vibrated  by  Dubuat,  .         .  295 

Parabola,  path  of  projectile,  .         .        .       258 

described  while  rotating,       .         .267 

a  tachytrope,     .         .         .         .368 

described  by  law  of  gravitation,    .  379 


ALPHABETICAL   INDEX. 


495 


Parabola,  condition  of  description, 

431  1 

Paraboloid,  brachistochrone  on, 

336 

path  of  heavy  body  on. 

416  j 

Parallel  and  equal  forces, 

132 

Parallelopiped  of  translation. 

'' 

of  rotation, 

14 

of  forces, 

36 

of  moments, 

39 

of  rotation-area, 

434 

Parameter  of  potential,     .         .         .         . 

54 

Perpetual  motion  impossible  in  nature, 

31 

Pendulum,  simple, 

255 

in  a  resisting  medium. 

281 

seconds,  of  uncertain  length. 

313 

spherical, 

418 

spherical,  related  to  the  gyroscope 

446 

Perturbations,  methods  of,         .         .         . 

459 

Planetary  perturbations,  case  of,  .        463 

465 

Platinum  sphere  vibrated  by  Borda, 

296 

PoixsoT,  analysis  of  rotation. 

12 

relations  of  axis  of  rotation  and 

of  maximum  rotation-area, . 

436 

velocity  of  rotation  about  axis  of 

maximum  area,  •         .         .         . 

439 

on  the  rotation  of  a  solid,  . 

443 

Point,  equilibrium  of,         .         .         .         . 

128 

motion  of, 

245 

Poissox,  modification  of  Laplace's  equa- 

tion,    

48 

theorem  on  Legexdre's  functions 

100 

on    the    pendulum   in  a  resisting 

medium, 

286 

on  the  top,         .... 

450 

Pole  of  synchrone, 

373 

Potential, 

29 

of  gravitation,        .         .         .         . 

45 

relation  to  its  parameter,  . 

54 

of  spheroid, 

99 

of  equal  and  parallel  forces, 

132 

curve,  

407 

Power  defined, 

3 

law  of, 

163 

gained  or  lost  by  a  moving  line. 

259 

Pressure  upon  the  brachistochrone,  . 

329 

Principle  of  living  forces, 

163 

of  least  action,      .         .         .         . 

167 

of  last  multiplier, 

228 

Progression,  rotary, 

456 

Projectile,  path  of,         ...         .       410 

disturbed, 464 

Projections,  theory  of  orthographic,       .       481 
PuisiEUX  on  the  tautochrone, .         .        .  326 

Q. 
Quaternions  of  Hamilton  promise  a  new 

progress  to  analytic  mechanics,     .  476 


R. 


Reference,  notation  of  in  tliis  book,  . 

4 

Residuals  to  express  integral   of  central 

force, 

380 

Resisting  medium.     See  Medium. 

Resultant  defined, 

36 

vanishes  in  equilibrium  of  transla- 

tion,          

128 

Resultant-moment, 

39 

in  relation  to  rotation, 

130 

of  gravity  for  centre  of  gravity,     . 

133 

RiCCATi  on  central  force. 

379 

Rolling  of  solid, 

457 

Rotation,  analysis  of,     . 

12 

combined  with  translation,   . 

16 

instantaneous  axis  of, 

19 

tendency  to, 

40 

of  expansion,     .... 

120 

equilibrium  of,       . 

129 

of  line  upon  which  a  body  moves 

about  a  vertical  axis, . 

261 

motion  of, 

433 

of  a  solid  body,     .         .         .         . 

434 

Rotation-area, 

433 

in  a  resisting  medium,  . 

475 

s. 

Screw  motion  includes  that  of  all  solids,         19 

Seconds  pendulum,  of  uncertain  length,   .  313 

Sections,  conic.     See  Conic  Sections. 

Shell,  attraction  of  spherical,    .         .         .56 

attraction  of  Ciiaslesian,        .         58 

Chaslesian  ellipsoidal,       .         .     70 

Newtonian,    ....         70 

Sleep  of  the  top, 451 

Sliding  motion,      .         .         .         .         .457 

Solid  motion  analyzed,     .         .         .         .18 

rotation  of,         .         .         .         .434 


496 


ALPHABETICAL  INDEX. 


Solution  of  a  partial  differential  equation,  199 

of  continuity  in  law  of  resistance,  273 

Sphere,  attraction  of,        .         .         .         .57 

having  catenary  upon  its  surface,  157 
vibrated  as  a  pendulum,        .         .  294 
a  synchrone,      .         .         .         .374 

condition  of  descrijjtion,         .         .433 
Spheroid,  potential  of,  .         .         .         .         99 

which  is  almost  an  ellipsoid,  .         .110 
almost  a  sphere,         .         .         .       Ill 
Spiral  logarithmic  path  on  a  rotating  line,  251 
logarithmic  described  against  fric- 
tion,  .         .         .         .         .         .274 

logarithmic  a  tautochrone,         .       325 
a  brachistochrone,         .         .         .  331 
logarithmic  a  tachytrope,  .         .       365 
logarithmic  described  when    cen- 
tral force  is  inversely  proportion- 
al to  the  cube  of  distance   .         .379 
path  of  the  axis  of  a  soUd,  .       44  3 

Stability  of  the  funicular, .         .         .         .  135 

Stader,  special  eases  of  central  force,.       379 
central  force  inversely  proportion- 
al to  the  cube  of  the  distance,     .  385 
central  force  inversely  proportion- 
al to  the  fourth  power  of  the  dis- 
tance,   404 

central  force  inversely  proportion- 
al to  the  seventh  j^ower  of  the 

distance, 40G 

Straight  line,  attraction  of  infinite,        .         52 
rotating  uniformly,  with  body  mov- 
ing upon  it,         ...         .  249 
described  by  heavy  body, .         ,       255 
rotating  uniformly  about  vertical 
axis,    and   described    by    heavy 

body, 2G2 

rotating   uniformly  about  an   in- 
clined   axis,   and   described    by 
heavy  body,    .         .         .         .269 
a  tachytrope,         .         .         .         .365 
Superposition  of  small  oscillations,         .       470 
Surfaces  of  the  second  degree  homofoeal,       79 

orthogonal, 79 

of  distorted  expansion,       .         .       119 
of  revolution  containing  catenary,  143 


Surfaces  of  revolution   containing   tauto- 
chrone,        322 

of  revolution  containing  brachis- 
tochrone,   .....  334 

with  point  moving  upon  it,         .       376 
Synchrone,        .         .         .         .         .         .373 

Syntachyd,    .         .         .      '  .         .         ,3  75 
Systems  of  integrals,  ....  203 

motions  of,         ...         .       458 

motions  in  resisting  medium,  .        .472 


Tachistotrope,  ..... 
Tachytrope,  ..... 
Tautobaryd,      ..... 
Tautochrone,         .... 

compound,     .... 

in  Lagrange's  form, 

restricted  by  Fontaine, 
Tension  of  the  catenary. 
Time  disturbed  in  Hansen's  method. 
Top,  spinning  of,  . 
Translation,  analysis  of,    . 

combined  with  rotation,    . 

tendency  to,  ... 

etjuilibrium  of,  . 

motion  of,      .... 

in  a  I'esisting  medium, 
Trifolia  of  Stader, 
Trajectory  of  level  surfaces,  . 


.  369 

364 
.  370 

316 
.  358 

359 
.  362 

139 
.  465 

449 

7 

16 

.     37 

127 
.  241 

474 
.  379 

32 


Variation  of  the  characteristic  function,    .  166 
of  a  function  of  the  elements  of  a 
determinant,   .         .         .         .180 

rotary, 456 

of  arbitrary  constants,        .         .       459 

Velocity, 3 

Vieille  on  the  motion  of  a  body  along  a 

rotating  straiglit  line,      .         .       262 
Virtual  velocities,  principle  of,  .        .         .7 

W. 

Weights,  French,  adopted,        .         .         ,  293 
Wooden  sphere  vibrated  by  Newton,        293 


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